The diffusion of single n-butane and n-decane molecules on a model stepped surface, Pt655, and on a corresponding flat surface, Pt111, is investigated using molecular-dynamics simulations and anisotropic united atom model. The surface step on Pt655 causes the alkane molecules to adsorb on the lower terrace in all-trans conformations with their long molecular axes adjacent and parallel to the step edge, and to diffuse anisotropically along the surface step via a constant wiggly motion without rotation or marked deviation from the parallel adsorption configuration. At relatively high temperatures, the alkane molecules can temporarily break away from the step edge but cannot migrate across the step edge in either the downstair or upstair direction. In comparison with the diffusion on Pt111, the diffusivity of n-decane is reduced by the surface step but its diffusion barrier is hardly affected. In the case of the shorter n-butane, however, the surface step significantly reduces the diffusion energy barrier and gives rise to higher diffusion coefficients at lower temperatures. Important implications of the simulation results are discussed.