So lets take a look at a few specifics...
Jesus Himself set aside, as we have seen, the food laws laid down in the Law. He additionally set aside much of the Law pertaining to ceremonial cleanliness; He routinely came into contact with those who were considered unclean, and proclaimed that His cleanness made them clean, not the other way around.
He did, however, affirm much of the Law in its dealings with human ethical behavior surrounding economic, sexual, and interpersonal behavior. In fact Jesus took these portions of the Law and made them much more deeply binding on His followers, precisely because the issue is not obedience to the Law, but learning to become the kind of people who would fulfill the Law without trying to; who would not murder, cheat, or steal, regardless of the existence of the Law.
So what do we do with new claims to do with the Law what Jesus did?
Do we allow Jesus' innovation because He is Jesus? Do we ban others' innovation because they are not Jesus? Or are we each and every one of us expected to do what Jesus did with respect to the Law?
A perfect example is the sexual ethic. In recent decades we have heard very unique voices within the Church. Throughout the history of Christian thought the sexual ethic has been quite homogenous: sexual pleasure is, and should remain, a culmination and a reflection of the vow to lifelong partnership between a husband and a wife. Throughout history there have been dissenting voices from outside the Church, however, we now see a growing minority within the church advocating that we set aside this clear consensus of Christian teaching on sexuality, and that we do so for precisely the reasons (according to these voices) that Jesus encouraged us to set aside certain portions of the law in His day.
I disagree with these voices, and here is why.
1) The traditional sexual ethic is consistent with agape love, the love that has grown up under the tutelage of the Law, and then come out into its own maturity. The inconsistency claimed by these advocates (if we love people we will allow them to express their love, and so we must change the sexual ethic) is based on a notion of love that is rooted in popular culture and has grown up under the tutelage of that culture, in many ways in opposition to the wisdom offered by the Law.
2) While it is my contention that God desires for us to come to a place of maturity wherein we are able to judge for ourselves where 'Love fulfills the Law' prudence demands that we recognize the primacy of Jesus' judgment! His words should carry more weight than our own. What seems wise to our minds must be set aside whenever it contradicts His wisdom.
We Christians have disparaged the Law, in some ways this is consistent with New Testament teachings, but in other ways we have gone too far. The Law is not evil, nor is keeping it evil; quite the opposite, the Law is good, and keeping it is what God desires. The evil is in us. It is in how we approach the Law, how we use it, or more precisely how we misuse and abuse it. The Law was never intended as a means for making men righteous, but rather as boundary markers pointing towards righteousness. The Law is God's tool, and we must use it consistent with His purposes for it. As the Messiah's people it still has a function for us, albeit muted since the coming of the Messiah. It serves for us, the purpose of providing the backdrop to Jesus' teachings. Like students who occasionally read through their old Algebra text to be clear about what it is that their Calculus professor is teaching, or read through their old American History texts to gain understanding into contemporary affairs.