The dynamics of Brownian particles diffusing across a one-dimensional, incoherent stochastic potential of mean force in the Smoluchowski regime has been intensely investigated by several groups. In recent work, we have developed a phenomenological equation of motion that extends this representation throughout the friction regime and, in particular, extends it to the low-friction regime relevant to surface diffusion. Resonant activation is observed throughout; it is manifested by a peak in the transport as a function of the correlation time in the potential fluctuations. The phenomenological equation of motion has now been utilized to probe the dynamics on a variety of one- and two-dimensional surfaces in order to provide a qualitative description of the fundamental factors that govern the surface hopping events of an adsorbate weakly bound to a metal surface. The primary focus is placed on differences that may arise when the substrate is modeled using a one-or two-dimensional potential of mean force, thereafter the effects of spatially coherent or incoherent barrier heights are also addressed. The two-dimensional behavior can be adequately described by the direct product of two separable one-dimensional analogues, as might naively be expected, provided the lattice spacing is sufficient to decouple the two degrees of freedom. Coherency between the barriers affects the rates to a smaller degree and is significant only when the barriers are strongly correlated in time.