Motions of individual particles within the stripe and square patterns formed in oscillated granular media are studied using numerical simulations. Our event-driven molecular dynamics simulations yield standing wave patterns in good accord with those observed in experiments at the same frequency and acceleration amplitude. The patterns are subharmonic and so return to their initial macroscopic state after two external cycles. However, simulations reveal that individual particles do not return to their initial position. In addition to diffusive motion, an organized flow of particles within the patterns is found; associated with each peak and each valley of the pattern is a pair of counterrotating convection rolls. The diffusion is anisotropic: transport perpendicular to stripes is enhanced over that parallel to stripes. This enhancement is computed as a function of the layer depth, acceleration amplitude, frequency, and coefficient of restitution of the particles, and is attributed to the effect of the advective motion. Velocity distributions, granular temperature, and the dependence of the diffusion coefficient parallel to the stripes on the average granular temperature are studied.