We present results from molecular dynamics simulations for the anisotropic self-diffusion tensor and the velocity autocorrelation functions of monodisperse systems of dense linear chain molecules under flow. Two molecular models are used in these simulations: The finitely extensible nonlinear elastic chain and the freely jointed tangent sphere chain. Nonequilibrium molecular dynamics is used to simulate these systems under planar Couette flow and planar extensional flow. Under planar extensional flow, results presented here are the first, from simulation, for diffusion and velocity autocorrelation functions of molecules, while for planar Couette flow, we compare the broadest range of conditions. An explicit derivation is provided of the Green-Kubo expression for the diffusion tensor. This expression is then used to derive the relation involving the mean-squared displacement-an often used alternative method to calculate diffusion coefficients. Velocity autocorrelation functions have been used, in combination with results on the alignment of molecules from a previous paper, to provide some details of the molecular scale dynamics that influence diffusive transport under flow.