Question: I have a theological question, completely unrelated to your post. I was reading this passage in the Bible: “Do not think that I came to destroy the Law or the Prophets. I did not come to destroy but to fulfill. For assuredly, I say to you, till heaven and earth pass away, one jot or one tittle will by no means pass from the law till all is fulfilled. Whoever therefore breaks one of the least of these commandments, and teaches men so, shall be called least in the kingdom of heaven; but whoever does and teaches them, he shall be called great in the kingdom of heaven.” Why don’t Christians follow the same 613 Commandments of the Orthodox Jews? How much of Jewish law did Jesus say should no longer be followed? Or did he?
Answer: This is an excellent question and deserves a carefully thought out answer. First of all, when Jesus said that he came not to destroy the law, but to fulfill the law, I believe He was saying that through His life and teachings, He represented everything the law was intended to convey. When He said, "Whoever therefore breaks one of the least of these commandments, and teaches men so, shall be called least in the kingdom of Heaven," He was referring to the moral teachings in the Sermon on the Mount, which was His interpretation of the law.
Also I believe He was saying that His sacrificial death on the cross would fulfill the types and shadows that were represented in the Jewish sacrificial system. In the Law, God required the blood of innocent animals to atone for the sins of the Jewish people. (Lev 17:11) Interestingly, Jesus was crucified on the day of Passover and his bones were not broken, fulfilling exactly the type and shadow of the Jewish Passover in the Old Testament. This is but one way that Jesus fulfilled the types and shadows in the Old Testament. There are many others and I would encourage you to read the New Testament events surrounding the death and resurrection of Jesus and compare them to the Jewish sacrificial system in the Torah. You are sure to find some surprises if you read the Torah with this in view.
As to your question of why Christians do not follow Orthodox Jewish law, the simple answer is, according to the New Testament, God no longer requires it from people any more. The law was never intended to actually make man righteous before God because it is impossible to follow it perfectly (Galatians 2:21,3:21). If one could follow the law perfectly (which is impossible) it would only lead to self righteousness, something that God condemns. Therefore, the Bible says, "By the deeds of the law no flesh shall be justified, for by the law comes the knowledge of sin." (Romans 3:20) The New Testament teaches that God has provided a way to justify all men by faith in Jesus Christ, apart from the works of the law (Galatians 2:16, Romans 3:28).
This does not mean that Christians are not under moral constraints. Here is the general rule in the New Testament. Everything in the law of Moses is abrogated except for that which the New Testament says still applies. I conclude this because the Bible says the law, which used to divide Jews and Gentiles, was abolished at the cross (Ephesians 2:14-16). The Apostle Paul also told the Roman Christians, "You are not under the law, but under grace." (Romans 6:14)
As far as I understand what is taught in the New Testament,the moral law as revealed in the 10 commandments, still applies. This does not mean that one is justified by how well he or she keeps the law, but it does say that those who make a genuine effort to keep the moral law have demonstrated true repentance. So, to answer your question as to which parts of the Jewish law are applicable for Christians today (or, I would add, for all people), as far as I know, each one of the 10 commandments is enforced in the New Testament except for the command to keep the Sabbath. Although most Christians worship on Sunday, the New Testament, in reference to the Sabbath, teaches to not be overly concerned about which days are more important than others. This would also include Jewish holidays and feasts (see Colossians 2:16). Although it is certainly not wrong to follow the Jewish customs of the Old Testament, as many messianic Jews do today, it is no longer required and can never be a substitute for genuine repentance toward God and faith in Jesus Christ. From the New Testament perspective, this applies to all people, Jews and Gentiles alike.
This is a longer answer than usual. I encourage you to look up the Scripture references I gave you to see if I am telling the truth. I hope this answer helps you along the way in your spiritual journey.