In fact, Jesus clearly upholds Tradition in the New Testament as we will see, but before we do that, let's look carefully at the scene that people use as evidence that Jesus is hostile to Tradition.

The entirety of it is that Jesus castigates the Pharisees for their corrupt traditions in a teaching recorded in Mark 7:1-23, and Matthew 15:1-20. The differences between the two accounts of this scene are negligible, and so I will go through the account in Mark:

Mark 7

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"Now when the Pharisees gathered together to [Jesus], with some of the scribes, who had come from Jerusalem,
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they saw that some of his disciples ate with hands defiled, that is, unwashed.
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(For the Pharisees, and all the Jews, do not eat unless they wash their hands, observing the tradition of the elders;
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and when they come from the market place, they do not eat unless they purify themselves; and there are many other traditions which they observe, the washing of cups and pots and vessels of bronze.)

[Already we see the type of tradition Jesus will be castigating. As Fr. William Most writes in "Commentary on the Gospels: The Thought of St. Matthew," "Here the Pharisees object that the disciples do not follow the oral law about washing hands. This law was later codified in the Mishnah (c. 200 A.D.), where a whole tractate, yadaim, dealt with washing the hands, and how much water must be used. If a man poured water over one hand with a single rinsing, his hand would be clean, but if he poured it over both hands with a single rinsing, it would be unclean unless he poured much more water (Yadaim 2:1)." There is nothing wrong with "rituals," of course (a wedding ceremony is a ritual), but the pharisaic laws regarding daily activities had become a kind of neurosis. Fr. William Most also writes, for example, "Thus the schools of Hillel and Shammai [two prominent rabbis] debated: if a hen lays an egg on a feast day, is one permitted to eat it - thus getting the fruit of illicit work. Hillel said no, Shammai said one may eat it."
Most gives another example: "Again, a group of Essenes in Jerusalem at that time noted the command of Dt. 23:12-14 to build the latrine outside the camp. They said now the camp is Jerusalem. So they built the latrine outside the city, a distance of 3000 cubits, more than one was permitted to walk on the Sabbath! (Cf. BAR Sept-Oct., 1984, p. 45)." To resume the scene:]

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And the Pharisees and the scribes asked him, 'Why do your disciples not live according to the tradition of the elders, but eat with hands defiled?'
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And he said to them, 'Well did Isaiah prophesy of you hypocrites, as it is written, 'This people honors me with their lips, but their heart is far from me;
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in vain do they worship me, teaching as doctrines the precepts of men.'
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You leave the commandment of God, and hold fast the tradition of men .' [Jesus is not condemning tradition as such, but "tradition of men."]
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And he said to them, 'You have a fine way of rejecting the commandment of God, in order to keep your tradition! [It is called "your tradition," which had become nonsensical and neurotic.]
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For Moses said, 'Honor your father and your mother'; and, 'He who speaks evil of father or mother, let him surely die';
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but you say, 'If a man tells his father or his mother, What you would have gained from me is Corban' (that is, given to God) --
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then you no longer permit him to do anything for his father or mother,
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thus making void the word of God through your tradition which you hand on. And many such things you do.'
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And he called the people to him again, and said to them, 'Hear me, all of you, and understand:

[Note what Jesus will now emphasize – what is wrong with this corrupt teaching, and He will say nothing about the idea of Tradition itself.]

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there is nothing outside a man which by going into him can defile him; but the things which come out of a man are what defile him.'
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And when he had entered the house, and left the people, his disciples asked him about the parable.
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And he said to them, 'Then are you also without understanding? Do you not see that whatever goes into a man from outside cannot defile him,
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since it enters, not his heart but his stomach, and so passes on?' (Thus he declared all foods clean.)
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And he said, 'What comes out of a man is what defiles a man.
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For from within, out of the heart of man, come evil thoughts, fornication, theft, murder, adultery,
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coveting, wickedness, deceit, licentiousness, envy, slander, pride, foolishness.
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All these evil things come from within, and they defile a man.'"

[That is how the scene ends, because that was the point of it: not, "Scripture versus Tradition" but "People, listen to reason: it's not the dirt on the outside that matters but the dirt in the soul."]

How Jesus Upholds Sacred Tradition

As we saw in a previous answer, the New Testament repeatedly teaches the value of Sacred Tradition (2 Thessalonians 2:15; 3:6; 1 Corinthians 11:2; 2 Timothy 1:13; 2:2; 1 Thessalonians 2:13; cf. John 20:30; 21:25; Mark 4:33-35; 6:34-35; Luke 24:27-28; Acts 1:3; extra-biblical Tradition alongside Old Testament accounts is used in Jude 1:9; 2 Timothy 3:8; James 5:17; 1 Corinthians 10:1-4; cf. Matthew 2:23; 23:2-4).

More important in this context, however, many of Jesus' teachings, too, are based in rabbinic tradition that does not appear in the Bible. Take the golden rule, for example. In Leviticus 19:18 it had the form of "You shall not take vengeance or bear any grudge against the sons of your own people, but you shall love your neighbor as yourself," but in the Gospels, Jesus teaches it in a different form:

"So whatever you wish that men would do to you, do so to them; for this is the law and the prophets."
(Matthew 7:12)

Jesus seems to have taken the form of this teaching from the work of Rabbi Hillel (40 B.C. – 10 A.D.), who taught, "What you do not like should be done to you, do not to your fellow; this is the whole Torah, all the rest is commentary." Talmud, Shabbat 31a, Nahum Norbert Glatzer, Hillel the Elder: The Emergence of Classical Judaism (New York: Schocken Books, 1966). To teach us, Jesus was free to use whatever sources He wanted, and if He could quote frequently from the Bible - which is the Word of God in the words of men - He could quote a rabbinic tradition like this just as easily. After all, Jesus is God who became man and lived in a place and time with certain true traditions that, as God, He could pick out and use infallibly. Though the golden rule had also appeared in Confucius, Buddhism, Lao-Tze, Hinduism, etc., it did not have so close a form: not only (a) do unto others as they would do unto you, but (b) this sums up the whole Bible. Hillel's writings were, of course, very respected in the Jewish community, and yet were a tradition apart from the Bible.

There are many other examples of this as well. Dorothy A. Lee is Professor of New Testament at the United Faculty of Theology and Dean of Chapel at Queens College, and the following is from her article published in the Journal of the Australian Council of Christians and Jews, Gesher (Bridge) Nov. 1996:

"Texts such as Hosea 6:6 — 'I desire mercy and not sacrifice'— are also found in Rabbinic writings (Matt 9:13). [That is, in Matthew 9:13, the words of Hosea 6:6 are quoted by Jesus, which was a text emphasized in rabbinic writings, and which suggests that tradition can help determine things in the Bible that are to be emphasized -- one of the functions of Tradition in the Catholic faith.] Also important for Matthew is the community's power of 'binding and loosing', a perplexing phrase that is also found in Rabbinic texts (Matt 16:19, 18:18)."

To take another example, according to Father William Most ("Commentary"), the Last Supper, too, probably has a close relationship with rabbinic theology at the time. Most writes,

"Jesus says His blood is poured out for the remission of sins. The wording is reminiscent of the blood of the Sinai covenant sacrifice in Exodus 24:5-8. That blood in Exodus splashed on the people signified that they were becoming as it were blood kinsmen of God. So He would be their goel the next of kin who, within the covenant bond hesed, had both the right and the duty to rescue a kinsman of his who was in dire difficulty. To drink His blood then was to be His kinsman, His brother, by a sort of blood transfusion. The Mishnah, Pesahim 10. 6 (from about 200 AD and so probably reflecting a tradition on hand at the time of Jesus) uses Exodus 24:8 to interpret the Passover wine as a metaphor for blood that seals a covenant between God and His people."

We also should not forget that Jesus wrote nothing down Himself (at least nothing that has come down to us). All his instructions were oral, and would presumably have followed the rabbinic custom of teaching disciples several statements that are committed to memory. Some such body of teachings must have been inculcated when Jesus sent out His disciples to teach during His ministry (e.g., Luke 9:1-6; 10:1-12), as well as when He gave His great commission after the resurrection (Matthew 28:19-20).

Finally, there is Jesus' famous upholding of the "Chair of Moses" as a teaching authority (at least for the time being), an element of the faith that does not appear in the Bible, but comes from tradition (Matthew 23:2-3).

To summarize, then,

1. The New Testament is filled to overflowing with outright references to the value of Tradition (2 Thessalonians 2:15; 3:6; 1 Corinthians 11:2; 2 Timothy 1:13; 2:2; 1 Thessalonians 2:13; cf. John 20:30; 21:25; Mark 4:33-35; 6:34-35; Luke 24:27-28; Acts 1:3; Tradition used in Jude 1:9; 2 Timothy 3:8; James 5:17; 1 Corinthians 10:1-4; Matthew 2:23)

2. Jesus Himself upholds Tradition, whether in simply calling it authoritative (Matthew 23:2-3; cf. 7:12), using what tradition has emphasized (Matthew 9:13), using the terminology of tradition (Matthew 16:19; 18:18), or in calling a particular tradition worthy of being followed (Matthew 23:23).

3. In castigating the Pharisees' corrupt tradition, Jesus was dealing with regulations that had clearly passed over into neurotic detail (such as not eating a particular egg because the hen had exerted itself on a feast day).

4. Jesus' own promulgation of His teaching would have been oral tradition: in all likelihood based on rabbinic models of memorization.

(Written by Christopher M. Butler)