I know this is prob. not going to be a very popular opinion here, but honestly, I have a hard time reconciling the two beliefs. If you feel you can reconcile the two beliefs, that's cool with me, and you are certainly not alone. Edgar Cayce was a famous example of a person who considered themselves Christian, as well as a believer in reincarnation. But I can't seem to reconcile the two.
As Inphanta said, the verse that said something about it being appointed unto men to die once, after which there would be judgement, is one example of a place in the Bible that seems to disagree with the compatibility. I don't know exactly how it goes, but think there's supposed to be something like that in there. Also, when Jesus was talking about being born again, wasn't there a man who asked how they could return to the womb? And didn't Jesus say it was a spiritual rebirth, not a fleshly one? I'll be honest, I'm not well versed in the Bible, so do your own research, but I seem to recall this story. Also, doesn't the Bible speak of hell as eternal? And if we are constantly reborn, until we reach perfection, and then we get to move on, how would we ever even reach hell? Unless this is hell, in which case, wouldn't that mean those of us who'd been here more than the first time had been sent to hell and would never break the rebirth cycle?
I just don't see how it's harmonious. I will say however, that there are a couple verses in the Bible that people cite as supporting reincarnation, that confuse me a bit, as well. Yet, given the above, I'm still not sure they are in harmony. But at the same time, that doesn't give people a right or a reason to treat you badly if you interprit things differently, and consider yourself a Christian who believes in reincarnation. I also live in a conservative area and admitting you believe in reincarnation would get you a lot of bad looks and rude comments. Some people are more accepting, less judgmental, etc., however.