We present an investigation of molecular permeation of gases through nanoporous graphene membranes via molecular dynamics simulations; four different gases are investigated, namely helium, hydrogen, nitrogen, and methane. We show that in addition to the direct (gas-kinetic) flux of molecules crossing from the bulk phase on one side of the graphene to the bulk phase on the other side, for gases that adsorb onto the graphene, significant contribution to the flux across the membrane comes from a surface mechanism by which molecules cross after being adsorbed onto the graphene surface. Our results quantify the relative contribution of the bulk and surface mechanisms and show that the direct flux can be described reasonably accurately using kinetic theory, provided the latter is appropriately modified assuming steric molecule-pore interactions, with gas molecules behaving as hard spheres of known kinetic diameters. The surface flux is negligible for gases that do not adsorb onto graphene (e.g., He and H2), while for gases that adsorb (e.g., CH4 and N2) it can be on the order of the direct flux or larger. Our results identify a nanopore geometry that is permeable to hydrogen and helium, is significantly less permeable to nitrogen, and is essentially impermeable to methane, thus validating previous suggestions that nanoporous graphene membranes can be used for gas separation. We also show that molecular permeation is strongly affected by pore functionalization; this observation may be sufficient to explain the large discrepancy between simulated and experimentally measured transport rates through nanoporous graphene membranes.