I don't believe there is a way to avoid it. It's how they keep track of the fact you "own" that app. If you ever need to d/l it again they won't charge you. I'm pretty sure that's how they handle the DRM for songs, books and videos you d/l too.
I do NOT have credit cards. Strictly cash and carry. I also had a problem, because just to register you need a credit card on Amazon. Annoyed me, because I would only be getting the "free" apps anyway!
What I did was used an old Visa gift card for the numbers. There was no money on it, and if I try to use it for a "pay" app, it wont work (as expected)....but it allows me access to that market.