Does every form of suffering arise from grasping?

The Buddhist idea that suffering arises from desire and that we can alleviate suffering by controlling our pions works to some extent, but there is still suffering that arises from disease, accidents and other situations that seem faultless to the sufferer. I understand that Karma and reincarnation are sometimes put into the picture in this case, but if one has no foreknowledge of one's wrongdoing's in a past life, doesn't the belief in something intangible and unproven go against the Buddhist ideals of using ones head? Or is there a Buddhist ideal of using one's head? This is where Buddhism falls flat for me. At the point where one has to have faith in something unproven, a past life, one has to still use faith and not one's own head. I'm pretty sure that one of the main points of Buddhism is that we ume our past lives wrongdoing. I've been stumbling over this one for awhile and would really like the perspective of a Buddhist on it.