By inhomogeneous I assume you mean disordered, i.e., a system with a noisy/random potential landscape.
I'm not sure which Mahan book you are referring to, however I found Akkermans and Montambaux' Mesoscopic Physics of Electrons and Photons to give a good discussion of the problem of wave propagation in disordered media.
Essentially the problem is "solved" by not trying to calculate the Green function for an individual system, but instead calculate averages for an ensemble of systems sharing similar characteristics, i.e., all being generated from the same random process. As long as the correlations in the inhomogeneity are spatially invariant, you restore a sort of general uniformity to the system and you can talk about a real conductivity, at least, in terms of the average conductivity. Also you can talk about average Green function (this decays rapidly, mind you), and correlation functions of conductivity.
In the end these ensemble average calculations end up being even more useful than directly computing Green functions for individual systems, since in experiment the disorder pattern is usually not known and so it would be impossible to make a direct comparison anyway.