The boundary-element method is used to solve Stokes equations for periodic arrays of force-free and torque-free rigid particles. Simple cubic arrays of spheres, spheroids, cubes, and clusters of spheres are subjected to a bulk simple shearing flow. The effective volume-averaged stress tensor for the suspension and the detailed velocity and stress fields throughout the Newtonian suspending fluid are calculated. We find that even crude meshes give very good volume-averaged results, but fine meshes are required to track local minima and maxima in the stress field. For simple cubic arrays of spheres, the boundary-element results are in excellent agreement with the analytical viscosity predictions of Nunan & Keller (1984). Even at the highest concentration of solids studied, no significant normal stress differences were observed, in agreement with Nunan & Keller's results (1984). Up to moderate concentrations of particles, the volume-averaged properties of the suspension display only a weak dependence on the particle geometry. Suspensions of spheroids and cubes behave approximately as suspensions of spheres on the average despite large differences in the local micromechanics of stress and velocity fields. Simple cubic arrays of clusters of spheres tend to behave on a macroscopic level as a cubic array of spheres whose effective volume fraction is about 150% of the total volume fraction of the spheres in the clusters.