We employ relativistic cosmological perturbation theory to study the evolution of linear peculiar velocities in Friedmann universes with nonzero spatial curvature and then compare our results with those obtained in models with flat spatial sections. Our study confirms earlier claims that the relativistic analysis supports peculiar motions considerably faster than those predicted by the Newtonian studies. We also find that the higher the matter density of the background universe, the faster the linear growth of the peculiar-velocity field. More specifically, velocity perturbations are found to grow faster in the closed Friedmann universes, than in their spatially open counterparts. Overall, our results seem to suggest faster bulk peculiar motions in overdense, rather than in underdense, regions of the universe.