If Jesus is the same yesterday, today and forever, and Jesus is God, that necessitates that the God of the New Testament is the same God as the God of the Old Testament. For many Christians, this is troubling. Does that mean we still have to be careful about what we eat, what we wear, what we do etc.? Does it mean we still have to severely punish certain sins? It's no secret that modern Christianity looks significantly different from Old Testament Judaism.
To put your minds at ease, I'm going to give the short answer first and say that no, we do not have to live as Old Testament Jews. This is not arbitrary. It's not "I don't like it, therefore it's not true." It's not "Society says this is barbaric, therefore it's allegorical." It's not "God suddenly changed His mind." The key to understanding the apparent "repeal" of the Old Testament is to understand its purpose.
First, let's give a Biblical analogy. In Romans 7:1-4, we read "Or do you not know, brethren (for I speak to those who know the law), that the law has dominion over a man as long as he lives? For the woman who has a husband is bound by the law to her husband as long as he lives. But if the husband dies, she is released from the law of her husband. So then if, while her husband lives, she marries another man, she will be called an adulteress; but if her husband dies, she is free from that law, so that she is no adulteress, though she has married another man. Therefore, my brethren, you also have become dead to the law through the body of Christ, that you may be married to another—to Him who was raised from the dead, that we should bear fruit to God."
Marriage is the perfect analogy here, because it is a circumstance that changes many things. Almost all Christians know that pre-marital sex is a sin. A man must remain a virgin until he is married, a woman must remain a virgin until after she is married. However, on the very same night the two are married and onwards, they may sleep with each other as often and as intimately as they want, and are even encouraged to do so to avoid temptation to commit adultery (1 Corinthians 7:5). The man did not change, the woman did not change. The circumstances, however, did. In the same way, sex outside of marriage is sin. A man may only have sex with his wife for as long as they both live, a woman may only have sex with her husband as long as they both live. Even after a divorce, Jesus says it is adultery in most scenarios to remarry (with exceptions for scenarios such as sexual immorality).
The law and Christ are both considered two different "husbands", and mankind is considered the same "wife". Before the resurrection, we were "married" to the law, the breaking of it being like adultery. Through Christ's death, the law was fulfilled (Matthew 5:17-18), passing away so that we are no longer bound to it. Now we are "married" to a new husband, Jesus. No husband is going to be particularly pleased if his new wife constantly longs for the man she once loved in place of him, so similarly Jesus does not require us to obey any laws that He did not renew in the New Testament.
But that's not to say the Old Testament is, in any way, invalid. We can't afford to just cut it out of the Bible and throw it in the bin. After all, it's still the word of the same God. It's just that for us living after the crucifixion, it has a different application. In Galatians 3:23-25, we read "But before faith came, we were kept under guard by the law, kept for the faith which would afterward be revealed. Therefore the law was our tutor to bring us to Christ, that we might be justified by faith. But after faith has come, we are no longer under a tutor."
Even though we need not keep the law, it still has a lot to teach us. First of all, it is a foundation for the New Testament. Without the Old Testament, we could not possibly understand the New. The New Testament, including when it quotes Jesus, repeatedly quotes the Old Testament. In John 5:45-47, Jesus even claims that Moses wrote about Him, saying if you don't believe Moses, you can't believe Him.
The Old Testament also shows us both the severity of sin and the extent of the mercy that covers it. The Old Testament is chock full of descriptions of punishments God would dole out for sin. That includes sins we see as "minor". Go to any atheist website and you will see non-believers complain about how easy God is to annoy. Many sins we see in our culture, such as homosexuality and consistent rebellion against parents, were punishable by death. Humans see this as an extreme punishment for a minor crime, but by reading from the words of our Creator, we see that there's no such thing as a "minor" crime, and that all crime carries a severe punishment. This punishment is what we were saved from.
Furthermore, we see just how far our sin took us from God in the Old Testament. While His love undoubtedly shines through, the priestly system of the Old Testament makes God seem very distant and very angry. Compare that with Hebrews 4 and it seems we're almost dealing with a completely different God. In reality, we're simply under a different covenant. Whereas the Jews needed to cover their sins with the blood of animals, Jesus' blood perfectly covers all of our sins, with the priesthood being a shadow of that.
With all of this in mind, we see that the Old Testament has not, as some say, been repealed or invalidated. One should study the Old Testament just as rigorously as the New, as neither can be correctly interpreted except in light of the other. However, that does not mean that we must obey every law in the Old Testament. Much of it, especially the ones most often picked up on by skeptics and Judaisers alike, has served its purpose. Christians are under a completely new covenant, and that should fill us with praise for the mighty God of both the Old and New Testaments.