I suggest, in brief, two sets of reasons not to embrace belief in reincarnation:
1. Even if one affirms some sort of numerical duality between brain and mind—a duality that need not involve any commitment to substance dualism—it still seems simplest to suppose that the brain gives rise to mental life. At least at first blush, this is what our experience and observation suggest. But, for reincarnation to make sense, one would need to imagine that not only a mind or soul numerically other than the brain but also a personal self with memories, personality, etc., exists in distinction from the brain. It will then be necessary to explain both (a) how this personal self comes into existence in the first place, if not as an initial product of brain activity and (b) how it comes to be intimately associated with a particular brain.
2. In tandem with these metaphysical or scientific objections to belief reincarnation, there are also existential objections. Most important is the devaluation of the empirical self, which seems likely to be swallowed up in some sort of transempirical self. Most people are interested in survival research, psi phenomena, etc., because of their hopes for themselves and their loved ones. But belief in reincarnation seems to take away with one hand what it gives with the other. The reincarnationist promises life after death. But the life after death offered by or in connection with belief in reincarnation is not the persistent life of the empirical self: the history, the relationships, etc., enjoyed by the empirical self seem to recede into unimportance, since they seem to play a limited role in constituting the contents of the actual self. This means, of course, that my own particular life seems less significant—but also that any and all empirical lives ultimately don't seem to matter a great deal, since the underlying self is evidently importantly distinct from them.
These objections are hardly decisive, but the first set suggests why I find belief reincarnation metaphysically baroque and the second why I find it existentially unappealing.