For a DM stack every line bundle has some power, which is the pullback of some line bundle on the coarse moduli space (assuming that our stack has a coarse moduli space). You can define ample, big and nef this way, I think. Then for example Kodaira vanishing holds, as far as I know. Here is an article in this subject: http://math.berkeley.edu/~molsson/KVrev2.pdf

For a DM stack (of finite type) every line bundle has some power, which is the pullback of some line bundle on the coarse moduli space (assuming that our stack has a coarse moduli space). You can define ample, big and nef this way, I think. Then for example Kodaira vanishing holds, as far as I know. Here is an article in this subject: http://math.berkeley.edu/~molsson/KVrev2.pdf