Surface diffusion of an adatom on a vicinal surface is investigated, using site-dependent hopping rates based on a model surface-potential profile of a regularly stepped surface. We solved analytically the coupled rate equations for the occupation probability of an adatom at a sufficiently long time, in analogy to the tight-binding theory of electronic structure. From this, the general relation between the hopping rates and the diffusion coefficient is derived. Formulas of both surface diffusion coefficients, parallel and perpendicular to a step edge direction, are obtained as functions of related atomic hopping rates at a terrace site, an upper edge site, and a lower edge site and of the step spacing. The fundamental mechanism determining the crucial role of step arrays on surface diffusion is clarified. No difference was found between step-up diffusion and step-down diffusion, even in the absence of inversion symmetry on the surface-potential profile. With Monte Carlo simulation, the effect of kink sites on surface diffusion is studied. Kinks greatly suppress the parallel diffusion coefficient, while they suppress only weakly the perpendicular diffusion coefficient.