A general diffusion process in a random medium associated with a random Dirichlet form with nonsmooth and nonbounded coefficients is considered, as drift transformation of a starting diffusion process in a random medium, with random infinitesimal generator in divergence form. The corresponding homogenization problem is studied and conditions are found such that the suitably rescaled original process converges weakly, as the scaling parameter is sent to the limit, to a diffusion process with constant coefficients. Stochastic analysis associated with Dirichlet forms is used in the proof.