A non-Markovian master equation is derived for the reduced probability density matrix of a subsystem interacting with a general reservoir of coupled anharmonic modes. Relaxation of the subsystem is mediated through resonant interactions with the reservoir. These interactions correspond to local vibrational motions with finite lifetimes due to diffusion/dephasing. The derivation assumes that the density of these interaction modes is very large so that they can be treated using statistical methods. The resulting master equation is shown to preserve probability, Hermiticity, and translational invariance, and conditions are stated which guarantee the positivity of the reduced density. At long time the master equation reduces to a Markovian equation of dynamical semigroup type. In the high temperature limit the theory is parameter free, and shows good agreement with the exact master equation of a spin-boson system.