I think it is helpful to remember that there are basic differences between the commutative and non-commutative settings, which can't be eliminated just by technical devices.

At a basic level, commuting operators on a finite-dimensional vector space can be simultaneously diagonalized [added: technically, I should say upper-triangularized,
but not let me not worry about this distinction here], but this is not true of non-commuting operators.  This already
suggests that one can't in any naive way define the spectrum of a non-commutative ring.
(Remember that all rings are morally rings of operators, and that the spectrum of a commutative ring has the same meaning as the [added: simultaneous] spectrum of a collection of commuting operators.)  

At a higher level, suppose that $M$ and $N$ are finitely generated modules over a
commutative ring $A$ such that $M\otimes_A N = 0$, then $Tor_i^A(M,N) = 0$ for
all $i$. If $A$ is non-commutative, this is no longer true in general.  This reflects the fact
that $M$ and $N$ no longer have well-defined supports on some concrete spectrum of $A$.
This is why localization is not possible (at least in any naive sense) in general in the non-commutative setting.  It is the same phenomenon as the uncertainty principle in quantum mechanics, and manifests itself in the same way: objects cannot be localized at points in the non-commutative setting.
 
These are genuine complexities that have to be confronted in any study of non-commutative geometry.  They are the same ones faced by beginning students when they first discover that in general matrices don't commute.  I would say that they are real, fascinating, and difficult, and people have put, and are currently putting, a lot of effort into understanding them.  But it is a far cry from just generalizing the statements in Hartshorne.