One of the big problems you run into is localization. Not all rings can be localized. You'll almost certainly need to restrict to rings satisfying the Ore Conditions. However, lots of natural rings do satisfy them, for instance "almost commutative" rings do. These are filtered rings whose associated graded ring is commutative. Among the rings of this form is the ring of linear differential operators on an open subset of a variety, and localization works out, so you get a sheaf, and you can look at modules, etc, they're called D-modules.
But anyway, the first few things to do are to decide on a class of rings where you can localize (or if you can't, you REALLY need to rework things from the bottom), and then you need to decide if you're looking at left, right or bimodules, including whether you're going to look at prime left/right/two-sided ideals, etc.
CAVEAT: I am a nonexpert in noncommutative AG, I just know a few places where the standard things in commutative AG break down a bit.
