We show, both heuristically and numerically, that three-dimensional periodic Lorentz gases-clouds of particles scattering off crystalline arrays of hard spheres-often exhibit normal diffusion, even when there are gaps through which particles can travel without ever colliding-i.e., when the system has an infinite horizon. This is the case provided that these gaps are not "too large," as measured by their dimension. The results are illustrated with simulations of a simple three-dimensional model having different types of diffusive regime and are then extended to higher-dimensional billiard models, which include hard-sphere fluids.