In fact, I revised some of problems in Chapter II Scheme theory in Hartshorne using Kontsevich-Rosenberg's machine. I have to notice, what you should deal with is probably module category over noncommutative ring. In noncommutative algebraic geometry, this is just category of quasi coherent sheaves on noncommutative affine schemes. But I did not restrict myself to noncommutative ring case. I try to do it in general noncommutative scheme, say a Grothendieck category or an abelian category. Noticed that, one can take grothendieck category as category of quasi coherent sheaves on quasi compact and quasi separated "would be scheme". So one should consider category of grothendieck category as category of "space" and morphism between spaces as iso class of inverse image functor. Rosenberg developed algebraic geometry in this 2-category. He introduced various spectrum for various destination. I should mention, spectrums for abelian category in his sense coincides with prime spectrum of a commutative ring when you take module category over commutative ring. In fact, one can define Zariski topology on this 2-categories using a family of conservative(faithful)exact localization functors(Serre subcategory in dual language). Then one can introduced the associated topology on the spectrum of abelian category. Then one can continue to introduce the "fiber" at each point of spectrum as stack of local category.(as fibered category) This is called geometric realization of an abelian category or Grothendieck category. Then one can take category of quasi coherent sheaves on this fibered category. At last, we get reconstruction theorem for noncommtative scheme. If we take the original category as quasi coherent sheaves of quasi compact(or not in general)quasi separated commutative scheme. Then we get reconstruction theorem for commutative scheme which means commutative algebraic geometry can be fully embedded into noncommutative algebraic geometry.
Because of this "Justify" theorem, we can develop various notions correspondent to commutative algebraic geometry. One can define noncommutative affine scheme (can be seen as category with projective cogenerator, then by Gabriel cheating theorem, equivalent to a module category). One can also define affine morphism, open/closed immersion/coimmersion(for the motivation of representation theory) picard group, vector bundles
One can also define differential operators in abelian category, monoidal category(for motivation of representation theory of quantum group and math phy), in particular, Noncommutative D-modules on noncommutative space, in particular quantum D-module on quantized flag variety. (I think this is related to the problem mentioned by siegels).
As is well known to all, Beilinson Bernstein's framework aiming to representation theory lives in triangulated category. Actually, there is indeed whole abelian picture developed mainly by Rosenberg and Lunts-Rosenberg-Tanisaki later.
In fact, for most(I think it should be all) problems in Hartshorne (facts in commutative algebraic geometry)has correspondence version in noncommutative algebraic geometry (in particular, what you mentioned, noncommutative ring)
There is indeed noncommutative flat descent theory in Konstevich-Rosenberg's work. I think the more accurate name should be categorical flat descent theory(Beck's theorem)
One more comment: What I mentioned above is ONE framework they developed.(Mainly for representation theory). There is ANOTHER framework introduced by them base on presheave view point(proposed by Gabriel-Grothendieck). They develop algebraic geometry in this view point which is NOT equivalent to the CATEGORICAL GEOMETRY I mentioned above in general. They coincides in affine case and then go to completely different direction. The mainly motivation for this view point is from Konstevich, he wanted to consider noncommutative grassmannian which might be helpful in understanding M-theory in Physics. Along this direction, they define noncommutative algebraic space, stack (DM and Artin) and so on.
last comment: One guy mentioned above, the category of rings did not have good property as commutative ones. But I guess this is not a very big deal. Rosenberg define so called right exact category(say category of rings, category of affine schemes, category of vector bundles). He developed whole homological algebra in this settings and Universal algebraic k theory, algebraic cycles. chow group and so on
I am sorry I have to go back to work instead of typing here. There are various noncommutatve algebraic geometry. If you are dealing with projective scheme, you might be interested in work of Artin's School on NC projective geometry
Several other comments: we have the notion of locally noetherian category whose objects is generated by noetherian object. For example, category of quasi coherent sheaves on noetherian commutative scheme is a locally noetherian category. We can play game in this setting. Then we can get whole commutative algebraic geometry of noetherian scheme
