The doctrine of Original Sin provides at least some insight into this by stating that there are two kinds of evil: physical evil and moral evil. Moral evil is that which comes about from humans' sinful acts--rape, murder, etc. The suffering inherent in these is usually not difficult to trace. Physical evil, on the other hand, springs not from present humans' wrongs but from the intrinsically disordered state of the universe as it is, brought on by the first sin (whatever that was; Catholicism at least does not make the absurd claim that it was a pair of naked people eating bad fruit). One metaphor might be that sin is a horrible pollutant in the spiritual and natural ecosystem of the world--the poison enters at some fundamental level, then spreads, contaminating things all the way up the food chain. It is not, technically, a divine punishment, but a simple consequence of the introduction of sin. It is this physical evil that brings disease, natural disasters, accidents, etc. even to the undeserving.

Small reassurance, I know. It bites that we have to deal with problems left over from the dawn of humanity. But it is an explanation, and one that many anti-religious people don't even bother to look into. They just assume that because they have an argument against it, the religion must be flawed.