Reincarnation isn't something I believe in, in the sense that I see no evidence that it's real. If there was repeatable evidence for it, then I'd believe in it. That being said, let's have a little fun...
Imagine that reincarnation does exist as a method of "training" the soul. Each incarnation in flesh refines and improves the soul a bit each time. Let's further presume that the idea is to improve sufficiently to the point that you can move to the "next level" (is this starting to sound familiar?). The question then:
At what periods of time in the known history of mankind would the soul have gotten the best education. Or in other words, in which period of history would the soul have gotten the most out of the least number of lives spent in flesh on Earth?
Before you answer quickly, it's important to realize that we have no idea what the criteria for judgement would be. What may seem like a wise and experienced soul (if indeed you could describe a soul that way without the resume of a life attached) may in reality be a naive soul.