I haven't spent much time contemplating reincarnation as real or not real. Here are my thoughts on the matter so far. My philosophical outlook is closest to Zen Buddhist and Taoist views, with some modifications.
I'll work from the assumption that we are entirely natural entities. This assumes also that our thoughts and souls/spirits/ghost whatever you want to name them are also natural. Given these assumptions and the experimental indications of undetectable dark energy and dark matter in physics, I posit that thoughts and soul are as yet undetected forms of energy. Our current scientific generalisations (Asimov calls them this; he figures "laws of nature" is a misnomer), say that energy can be neither created nor destroyed. So what we have is an energy form that creates thoughts and can influence the physical structure of the brain (neurplasticity experiments show that visualisation can produce a measurable change in neural mapping). It is not unreasonable to assume that such an energy pattern (soul) could retain pattern coherence after the death of the physical body. If I take this assumption further it is plausible that the energy pattern that is soul could then "reattach" itself to another bioligical body of similar configuration to its last one. Humans, after all, have the vast majority of our DNA code in common with each other and other mammals. I believe the difference between us and dogs is something like 2% and less than 1% between us and chimpanzees. Still we have a huge ammount of genetic variance between individuals.
So, though I haven't really made any stated belief or disbelief in reincarnation, I accept it as a naturally possible event.
It is not relevant to my task of being a moral human in this life. I will find out for certain once I die. If there is reincarnation and the soul I'll know, and if there is just nothingness then I won't ever be disappointed. ;-)