We present a molecular dynamics study of the motion of cylindrical polymer droplets on striped surfaces. We first consider the equilibrium properties of droplets on different surfaces, we show that for small stripes the Cassie-Baxter equation gives a good approximation of the equilibrium contact angle. As the stripe width becomes nonnegligible compared to the dimension of the droplet, it has to deform significantly to minimize its free energy; this results in a smaller value of the contact angle than the continuum model predicts. We then evaluate the slip length and thus the damping coefficient as a function of the stripe width. For very small stripes, the heterogeneous surface behaves as an effective surface, with the same damping as a homogeneous surface with the same contact angle. However, as the stripe width increases, damping at the surface increases until reaching a plateau. Afterwards, we study the dynamics of droplets under a bulk force. We show that if the stripes are large enough the droplets are pinned until a critical force. The critical force increases linearly with stripe width. For large enough forces, the average velocity increases linearly with the force, we show that it can then be predicted by a model depending only on droplet size, contact angle, viscosity, and slip length. We show that the velocity of the droplet varies sinusoidally as a function of its position on the substrate. However, for bulk forces just above the depinning force we observe a characteristic stick-slip motion, with successive pinnings and depinnings.