Using Stokesian dynamics simulations, we examine the flow of a monodisperse, neutrally buoyant, homogeneous suspension of non-Brownian solid spheres in simple shear, starting from a large number of independent hard-sphere distributions and ensemble averaging the results. We construct a novel method for computing the gradient diffusivity via simulations on a homogeneous suspension and, although our results are only approximate due to the small number of particles used in the simulations, we present here the first values of this important parameter, both along and normal to the plane of shear, to be obtained directly either experimentally or numerically. We show furthermore that, although the system of equations describing the particle motions is deterministic, the particle displacements in the two directions normal to the bulk flow have Gaussian distributions with zero mean and a variance which eventually grows linearly in time thereby establishing that the system of particles is diffusive. For particle concentrations up to 45%, we compute the corresponding tracer diffusivities both from the slope of the mean-square particle displacement and by integrating the corresponding velocity autocorrelations and find good agreement between the two sets of results.