The problem is that the category of (non necessarily commutative) rings does not have all the nice properties of the category of commutative rings. First and most obvious obstacle comes when ones tries to define the spectrum, since there are three different valid options for prime ideals (left, right or two sided); by picking two sided ideals one could define the Zariski topology just fine, but normally noncommutative rings don't have enough two-sided prime ideals to make the noncommutative spectrum interesting. Among other things it is in general not possible to reconstruct the ring out of the spectrum. As someone told me once "it doesn't matter how you define a point in a noncommutative space, there are never enough of them".
More subtle problems arise at the level of localization, on the hand one need to impose the Ore conditions, but even for rings that satisfy them one still has the problem that for noncommutative rings localization functors do not commute with each other. A possible detour around this problem was taken in the late seventies and early eighties (in what could be called the origin of noncommutative algebraic geometry) by Fred Van Oystaeyen, involving mainly replacing the naive notion of prime spectrum by more subtle ones (torsion spectrum, localization spectrum). A more recent summary of those viewpoints and their developments is in the book by Van Oystaeyen Algebraic Geometry for Associative Algebras.
Edit: After Kevin's clarification, a nice survey on the history and different approaches to noncommutative algebraic geometry can be found at the entry noncommutative algebraic geometry in nLab.
