Molecular simulations are carried out to address the structure and atomic diffusion at grain boundaries. We use an inherent structure approach, which maps each configuration in a molecular dynamics trajectory to the potential energy minimum ("inherent structure") it would reach by a steepest descent quench. Dynamics are then decomposed into a combination of displacements within an inherent structure and transitions between inherent structures. The inherent structure approach reveals a simple structural picture of the grain boundary that is normally obscured by the thermal motion. We apply our methodology to polycrystalline MgO. Grain boundary atoms are identified as atoms that are undercoordinated in the inherent structure, relative to those in the perfect crystal. Our method enables the calculation of grain boundary diffusion coefficients without arbitrary assumptions about which atoms or spatial regions belong to the grain boundary, and the results are shown to be consistent with estimates from experiments. The inherent structure approach also enables the elementary steps in the diffusion process to be elucidated. We show that the process in MgO grain boundaries primarily involves vacancy hops, but that there is also significant motion of other nearby atoms during such a hop.