This paper develops a multistep input-mapping data-driven model predictive control scheme for unknown constrained linear systems. Instead of identifying the model online or relying on costly offline experiments, the method maps current and future inputs to previously measured online input-state trajectories and expresses future states as combinations of past states. A moving data window continuously refreshes the information used by the controller. Two computationally efficient robust MPC formulations are derived with feasibility and stability guarantees, and simulations demonstrate improved performance and computational efficiency.