We present a new rate theory which accounts for anharmonicities (nonlinearities) in the solute potential (force) over the complete range of solvent damping. The theory is based on a new method for calculating energy diffusion rates which incorporates anharmonicity-induced solvent-solute reaction path curvature and is thus valid throughout the intermediate to large damping regimes. This energy diffusion factor is combined with the microcanonical variational transition state theory spatial diffusion correction factor. The new theory is applied to the case of a cubic solute potential coupled to a long time scale bath and shown to be significantly more reliable than the turnover theory of Pollak, Grabert, and Hanggi [J. Chem. Phys. 91, 4073 (1989)] in the Kramers turnover regime.