I don't see how reincarnation is any less plausible than any other descriptions of the afterlife.
Your main objection seems to be that there's no way of knowing that reincarnation is taking place, because you never have any memories of your previous lives. Well, some spiritual sects believe you actually do have access to these memories. Maybe it's just that memories from previous lives are "filtered out" by the immediacy of your current life. Maybe when you realize brahma, you will finally remember all of your lives (maybe when you get to heaven, you'll have a perspective on your own spirituality and and understanding of God that you never could have had as a mortal human). Furthermore, surely the gods know about your past lives; they could reveal it to others via prophets, and it's plausible that shamans could be able to tell you about these lives (you're not going to say that you don't believe in prophecy, are you?)
So basically I don't see any serious epistemological objection to reincarnation that isn't an equally valid argument against heaven and hell. The only people who ever went to heaven or hell came back to tell us about it were Jesus and Dante. I'm sure Hinduism has its share of prophets who told the people all about reincarnation. Also, it's not clear to me that most Christian sects take Dante's claims literally.
I think you guys are left with a perhaps more materialist and hence more primitive reflection of your own faiths.
As for reincarnation in Hinduism, Wikipedia claims it has been around for at least two thousand years, possibly up to three thousand years.
-Erasmus