The discrete element method allows the simulation of complex behavior of granular materials without constitutive laws. While in two dimensions shape-effects are well established, in three dimensions there is no universally applicable simulation algorithm for non-spherical particles. We will first present a force model for convex polyhedral particles, using the “overlap” of non-deformed polyhedra as a “measure” of the elastic force and explain the overlap computation algorithm. With this elastic force model, we then show that the continuum-mechanical sound velocity can be recovered for a space-filling packing of cubic blocks. Further, we show simulation results for heaps for which we obtain realistic high angles of repose, which corroborates the reliability of our simulation method and which further more shows that with our method, a larger phenomenology is accessible than with round particles.