We study the nature of non-Fickian particle transport in 3-D porous media by simulating fluid flow in the intricate pore space of real rock. We solve the full Navier-Stokes equations at the same resolution as the 3-D micro-CT (computed tomography) image of the rock sample and simulate particle transport along the streamlines of the velocity field. We find that transport at the pore scale is markedly anomalous: longitudinal spreading is superdiffusive, while transverse spreading is subdiffusive. We demonstrate that this anomalous behavior originates from the intermittent structure of the velocity field at the pore scale, which in turn emanates from the interplay between velocity heterogeneity and velocity correlation. Finally, we propose a continuous time random walk model that honors this intermittent structure at the pore scale and captures the anomalous 3-D transport behavior at the macroscale.