One that I heard recently, which I think can actually be a fairly common thought, goes something like this: "True Christians are supposed to throw out all of the Old Testament anyway, because Jesus said we're done with all of that." Essentially, saying that the Old Testament is no more than some interesting backstory at best, and has no relevance to how Christians are supposed to live today. There's a few reasons that I think this doesn't hold water, but firstly I'll have a look at why some people could see this as a legitimate idea.
Who It Was Written For
The Old Testament was written by, and for, the Israelite people. This is clearly shown throughout the books within it; and, as such, there are many references that only make sense in the time period. I talked about a few of those here a while back. There are lots of genealogies, lists of people, counts of things being sacrificed, rules and regulations specific to the Israelite people - none of which would have much immediate impact on our life today.
Jesus spoke about being the fulfilment of the Law (the Old Testament was commonly referred to in the Bible as the Law and the Prophets; the Law is specifically the first five books, also known as the Torah or Pentateuch); and many would argue that if something is fulfilled, it is no longer needed. There are also many times that Jesus seems to act in contrast to the Law; well-known examples include him healing people on the Sabbath, as well as associating with people that were 'unclean' - particularly lepers.
There are specific times when Paul goes against the words of the Old Testament; for example, he says that new Christians do not have to practice circumcision. In one of his letters, Paul writes, "By calling this covenant “new,” he has made the first one obsolete; and what is obsolete and outdated will soon disappear." (Hebrews 8:13) This would seem to be saying exactly what this argument is putting forward.
Now, let's have a look at the other side.
Jesus' Words"Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfil them." These are the words that Jesus spoke, in Matthew 5:17. And a number of other times in the Gospels, Jesus references the Old Testament; speaks about the importance of what was written there; and says that, "...until heaven and earth disappear, not the smallest letter, not the least stroke of a pen, will by any means disappear from the Law until everything is accomplished." That seems to be fairly definitive. But let's keep looking, shall we?
Paul's WordsIn 2 Timothy 3:16-17, Paul writes: "All scripture is given by inspiration of God, and is profitable for doctrine, for reproof, for correction, for instruction in righteousness: That the man of God may be perfect, throughly furnished unto all good works." Here, Paul is talking about Old Testament scripture - the New Testament hadn't been written yet! And while there are many times that we can find Paul talking about Christians no longer being under the Law, clearly he still believes that the Old Testament is very important for Christian living.
Purpose Of The Old Testament
In the New Testament, both Jesus and Paul talk about the Law and the Prophets acting to point towards Jesus; they are the foreshadowing of what is to come. As such, the Old Testament can tell us about who Jesus is, and help us to see the reason that he came. The Old Testament also contains many standards for living that are still helpful to strive for today; the Ten Commandments are a good example of this.
So, we have a bit happening here from both sides. Paul writes about not being under the Law; yet talks about the benefits of Scripture. Jesus talks about fulfilling the Law; yet not being here to abolish it. And there are many things in the Old Testament that seem extraneous; yet many can also apply to our lives today. So where do we stand?
In the end, unfortunately, it gets a bit muddy. However, two things that I think can be definitively said are; Christians are not under the Law, but the Old Testament should not be thrown out. It still has relevance and importance today for people's lives. As to anything more specific than that - you'd probably have to go to a proper Bible scholar. But that's my look at it for now. Hopefully I'll be doing this more often over the next while!