Wow, there are a lot of replies to this already, and it seems like lots of them hit my "must reply ..." button. 

Aniese of Learsi wrote:
Death (Biblical death, as in the separation of the soul from the body) occurs once, and there is no coming back, in any way, shape or form. This is vitally important because of what Christ did for us. If there is a way to defeat death without Him, what is the point?
This is true in principle, but ... 
1. We have several examples in the Bible of dead people coming back to life, with Lazarus as the most notable example (though there's a famous science fiction novel whose title I'm drawing a blank on, in the after-the-atomic-wars subgenre, that, I'm told, includes Lazarus as an unwillingly immortal character because "it is given to man to die once ..." and he's used up his one death already) but going back at least to the time of Elisha (2 Kings 13:20-21). 
2. And also, strictly speaking, "reincarnation" is not "defeat[ing] death", as the new body will 
also die. The part of the "it is given to man to die once ..." passage that it arguably contradicts is the next phrase: "... and afterwards to face judgment."
3. And, on the gripping hand, 
physical resurrection and eventual immortality is but the least part of what Christ's work does for us. Without it, we are 
already dead, 
spiritually dead, and salvation makes us spiritually alive and begins to bring us to 
full spiritual life.
Cassandra wrote:
What exactly is bad about reincarnation, beyond blurring the lines and being obviously impossible?
The trouble is that, apparently, it's not "obviously" impossible, because there are millions---no, 
billions of people who would insist that it's the way the world works.
Varon wrote:
Or just returning from Avalon.
Depending on what "Avalon" means, that may actually 
be "reincarnation". If "Avalon" is the name for a place like "Elfland" or "Underhill" where time goes all ... wonky ... and you can sleep for a night and wake up a hundred years later or live for fifty years and step out with only a week having passed, then yes, that's usefully distinct from reincarnation. But if you mean the physical island (which I believe actually exists ... though can any British HWers correct my extremely rough geography on that point?) on which he was (most of the legends say) buried, his body is almost certainly entirely rotted by now, so our two choices are resurrection ("raising up again") and reincarnation ("re-bodying" or "re-en-fleshing").
I agree with Sir Emeth wholeheartedly, except that I did not know about elvish reincarnation in Tolkien's Middle Earth. And I find his three-world system 
fascinating. (By the way ... tangentially ... what's the proper adjective form of "Ithelak-Alronia"?)
Sir Emeth Mimetes wrote:
Now that is a cool idea... someone is born with the entire memory of someone who died centuries ago, even though they are completely individual people.
Whether that's even 
possible (logically possible, i.e. conceivable, I mean) depends on the nature of personhood and identity (in what they consist), which is one of the "enduring questions" we looked at in my (college) freshman philosophy course.
In my own work, I have a few more-or-less-minor details that are arguably forms of reincarnation. First is that the Vaynar (angel-equivalents) cannot be permanently killed until their last God-appointed task is finished. They (that is, their bodies) can be killed (which is in practice not-quite-impossible anyway), but they usually appear instantly somewhere else (at full strength). Second, more worryingly (from the positions taken above), in the bits of my Shine Cycle that take place in our universe, I want to do a lot of legends-based-somewhat-in-fact worldbuilding flourishes, with some of the characters having appeared in different legends under different names (like Helen of Troy, Rhiannon ferch Hyfedd, and Guinevere being the same person) and then turning up later. This would seem to occasionally require someone (usually someone not-quite-entirely-human) to be re-incarnated ... which is, as we've concluded, an at least potential problem, depending on how it's handled.