In this work, we investigate the dynamic advancing and receding contact angles, and the mechanisms of motion of water droplets moving across nanopillared superhydrophobic surfaces using molecular-dynamics simulation. We obtain equilibrium Cassie states of droplets on nanopillared surfaces with different pillar heights, groove widths, and intrinsic contact angles. We quantitatively evaluate the dynamic advancing and receding contact angles along the advancing direction of an applied body force, and find that they depend on the roughness parameters and the applied body force in a predictable way. The maximum dynamic advancing contact angle is 180, and the minimum dynamic advancing contact angle is close to the static contact angle. On the receding side, the maximum dynamic receding contact angle is as large as 180, while the minimum dynamic receding contact angle is close to the intrinsic contact angle of smooth surface. Interestingly, water droplets exhibit a "rolling" mechanism as they move across the surface, which is confirmed by movies of interfacial water molecules, as well as droplet velocity profiles.