We develop a microscopic theory of surface diffusion of adatoms using the Mori continued-fraction formalism. Along the surface plane, the adatom motion is extended leading to diffusive behavior, while motion perpendicular to the plane is assumed bounded and oscillatory. In the high-friction limit, we find a novel analytic solution for the diffusion tensor in terms of generalized adiabatic potentials. We show how the inclusion of vertical motion can cause large quantitative changes in the values of the diffusion coefficients, while keeping the universal properties of surface diffusion in the high- and low-temperature limits qualitatively unaltered. We explicitly compute the diffusion tensor for a variety of different lattices and potentials. In the high-temperature limit, the theory recovers the diffusion of a Brownian particle in a viscous medium. In the low-temperature limit, we demonstrate how the Arrhenius form of activated diffusion and the geometric random-walk form of diffusion anisotropy arise from the theory.