First of all, I'd like to say I'm a scientist. This means fact will always come before faith, even if it hurts. In general, one's opinion doesn't mean much if the question itself has no meaning, but when the question is well phrased and has a quantitative background, then experiment is the ultimate - and decisive - destination.
Secondly, I'm afraid this will be a rather long question.
According to Walpola Rahula in his book What the Buddha Taught, Buddhism denies the existence of a soul or spirit:
Buddhism stands unique in the history of human thought in denying the existence of such a Soul, Self, or Atman. According to the teaching of the Buddha, the idea of self is an imaginary, false belief which has no corresponding reality, and it produces harmful thoughts of 'me' and 'mine', selfish desire, craving, attachment, hatred, ill-will, conceit, pride, egoism, and other defilements, impurities and problems. It is the source of all the troubles in the world from personal conflicts to wars between nations. In short, to this false view can be traced all the evil in the world.
This pleases me very much. As a scientist, I cannot conceive the existence of something we cannot observe or measure that lives on after I die. Everything Buddhism talks about pleases me very much: everything is dukkha, there's a way out of dukkha, we are unhappy because we cling to everything, and the path outside this unhappiness is to understand nothing has intrinsic value or meaning. In resume: I think Buddhism, unlike the other religions I've been exposed to, is beautiful and makes sense. But then I have a problem - and it's a big one, because it's in the eye of the tornado.
According to Walpola Rahula (same book),
Will, volition, desire, thirst to exist, to continue, to become more and more, is a tremendous force that moves whole lives, whole existences, that even moves the whole world. This is the greatest force, the greatest energy in the world. According to Buddhism, this force does not stop with the non-functioning of the body, which is death; but it continues manifesting itself in another form, producing re-existence which is called rebirth.
Now we have a problem. I've seen people discussing the differences between mind, brain, consciousness, etc. Well, science is pretty clear: the evidence is now overwhelming that every aspect of the mind is produced by the brain. I'm pretty familiar with NMR scanning and did read some hundreds of articles about the subject, and we have a huge, massive consensus that nothing survives our death. Not a force, not a will... Nothing. We can't see nothing. It doesn't exist.
Now, I have come to a more traditional (non-westernized) version of rebirth by reading this wonderful, majestically written article by an evolutionary biologist (and aspiring Buddhist): even though people like to talk about rebirth as a deterministic process, it is not. It appears Buddha himself (see article) has said that rebirth is like lighting a candle with another's fire: everything is lost and there cannot be a causal process that links you to "former lives". Actually, the only part of rebirth that doesn't violate scientific laws is to say that my atoms will be reborn in other creatures and objects. And that pleases me very much.
Now, a famous quote says that:
If science proves some belief of Buddhism is wrong, then Buddhism will have to change.
It is attributed to Dalai Lama (probably wrongly), and is found in this great article by Tenzin Gyatso, where many scientifically proved benefits of meditation are discussed. If this sentence makes sense to Buddhists, why do they keep insisting in the concept of rebirth? Science is clear: there's no support of any evidence in its favour, at least not in the way I see people treating the subject. Buddha's metaphor about "lighting candles" is much better to have as an example, since it doesn't contradict facts, even though it doesn't make a lot of sense either.
I'd therefore like to know passages from suttas where Buddha affirms that knowledge of past lives is unattainable, or at least where He elucidates reincarnation is not something causal. If there aren't any, why do people still believing in this concept? Only because they feel safer?
Edit: This question is turning into a chimaera of weird unwanted lessons: people telling me I'm not ready for knowledge, or that articles that elucidate their misconceptions are not valid, or that I don't know the meaning of truth, or people trying to teach me concepts my PhD is about, and even people saying their own views are perfect. Until now, the most useful references where given by Dhammadhatu, and the best answers by ChrisW and Tenzin Dorje. Please refrain from counselling and preaching.