I started looking into this when I went to Cambodia but it never went any further. Anyways, I think you can buy a used unlocked phone from local mobile stores that offer many phone plans (not stores owned by carriers). Then the best thing to do is buy sim cards at each country you enter. I did research the international sim cards but I thought they were expensive for minutes. I am not 100 % on my research but I believe the above would be the lowest cost way to have a phone. Boy, I am similar to you as I have a prepaid t-mobil cell phone (very basic-no games, ringtones, camera, data etc).
Hopefully this thread will help with other information to actually get a phone that is unlocked and takes sim cards.

Tony