I'm a bit late to this party, but as someone who started off not-beliveing in rebirth, then became agnostic about it, then firmly believed in the doctrine of rebirth, perhaps my comments could be of some value.
1. Even from a purely materialistic point of view, matter does not warp into and out of existence. Every atom in the Universe has existed at least since the start of the Universe. When someone dies, the mattter that makes up their body does not warp out of existence.
2. The mind(or inner experience) is immaterial. When I was agnostic about rebirth, the argument that the mind needed an immaterial cause, in addition to material causes, did not make sense to me. This is because I did not believe that the mind was purely immaterial. What eliminated my doubt, was reading the conversation between the Buddha and Ananda in the Shurangama sutra. In a nutshell, the Buddha states that the mind can not be found inside the body, outside the body, both inside and outside the body, or not inside the body and not outside the body. Since that makes the mind obviously immaterial, then it obviously needs an immaterial cause, in addition to material causes.
3. Since the material does not warp out of existence upon death, why would the Universe have special rules for the immaterial?
4. The exact mechanics for rebirth are a matter of dispute. Where I live now, Thailand, it is believed that rebirth is instantaneous. In Tibet however, they believe that there is an intermediate period before rebirth. I find both views are acceptable. The Buddha didn't teach "facts". He taught pragmatic truths that are useful for liberation, and nothing else. It's interesting that rebirth always seems to be what is brought out as "unscientific", but no one bats an eyebrow at matter being described within the system of Indian elements. Matter being described with Indian elements is sufficient for liberation, even though it may not be scientifically accurate.
5. Privately, being a nerd, I view what is reborn as being the remainder of an unbalanced equation. Balancing the equation results in Nirvana. Dogen sought to achieve balance, in this very life, by the practice of Shikantaza.