Diffusion of adsorbates on a square lattice containing steps every fourth row was investigated by Monte Carlo simulations. For the case where steps are attractive, i.e. step sites bind more tightly than terrace sites, there is very little diffusion anisotropy at low coverage, since steps act as traps about equally for diffusion parallel and perpendicular to steps. As coverage increases steps become highly populated and then act as barriers for diffusion across them so that anisotropy increases, particularly as temperature is lowered. Diffusion parallel to steps then becomes temperature independent and diffusion perpendicular to steps remains activated. For diffusion coefficients obtained by the fluctuation autocorrelation method the inability to leave terraces at high theta and low T introduces artifacts for measurements perpendicular to steps. For repulsive steps, constituting diffusion barriers but not serving as adsorption sites anisotropy is pronounced at all temperatures; diffusion parallel to steps is temperature independent, and diffusion perpendicular to steps is activated at all coverages.