Let $R$ be an $\mathbb{N}$-graded ring, and let $\Gamma : R\text{-gr.mod} \to R\text{-gr.mod}$ be the left exact functor defined by $$ \Gamma(M) = \{ m \in M \mid R_{\geq n} m = 0 \text{ for some } n \geq 1 \}. $$ The local cohomology modules (with respect to the irrelevant ideal $R_{\geq 1}$) of $M$ are defined to be $H^i_{+}(M) := R^i \Gamma(M)$. One can show that $H^i_+(M) \cong \lim_{n \to \infty} \text{Ext}^i_R(R/R_{\geq n}, M)$.
Under mild hypotheses, when $R$ is commutative, it is well-known that one can compute local cohomology by computing Cech cohomology. However, when $R$ is noncommutative, I have no idea how to compute local cohomology. In principle, I just need to compute an injective resolution, but for most rings it's difficult to explicitly write down any injective modules. Alternatively, I suppose I could calculate $\text{Ext}^i_R(R/R_{\geq n}, M)$ for all $n \geq 0$ and then take a direct limit, but this also seems difficult. So my question is:
Is there a general method for computing local cohomology over a "nice" noncommutative ring $R$?
Here, "nice" can be understood to mean noetherian and a domain/prime (which ensures the existence of a Goldie quotient ring, for example).