I'm not sure what you mean by "rings which look like D" but here's one point of view:
the de Rham functor is just derived Hom from the structure sheaf $O$, i.e. you're testing all D-modules against your favorite one. One can imagine an analog in any context where you have a favorite module. For D-modules with an integral twist, you can just use the corresponding line bundle instead of O, and again get a Riemann-Hilbert correspondence with constructible sheaves (this is a little silly though since all of these categories with integral twists are canonically equivalent).

More generally if your twist is a complex linear combination of line bundles (eg always in the analytic topology) you can think of twisted D-modules as monodromic D-modules: these are D-modules on the total space of a principal torus bundle, which are weakly torus equivariant, and have
some fixed monodromy along the fibers. For example in the case of the flag variety all sheaves of
twisted differential operators can be viewed this way using the "basic affine space" $G/N$.
So then you can apply the ordinary de Rham functor upstairs. This gives a Riemann-Hilbert correspondence with monodromic constructible sheaves on the total space, or if you prefer, with constructible sheaves on a gerbe over the base (I think this is addressed in another MO answer about complex powers of line bundles, I learned it from a paper of Kashiwara). The class of the gerbe in $H^2(X, C^\times)$ is just the exponential of the Chern class of the TDO, considered as a class in $H^2(X,C)$. Is that the kind of answer you're looking for?

D-Modules, Perverse Sheaves, and Representation Theory, Birkhauser, 1995 (see 11.2). Can you clarify how your question fits into the existing framework? (Also, a tag ag.algebraic-geometry might be useful.) $\endgroup$