In this paper, we propose a novel robust model predictive controller for tracking periodic signals for linear systems subject to bounded additive uncertainties based on nominal predictions and constraint tightening. The proposed controller joins optimal periodic trajectory planning and robust control for tracking in a single optimization problem and guarantees that the perturbed closed-loop system converges asymptotically to a neighborhood of an optimal reachable periodic trajectory while robustly satisfying the constraints. In addition, the closed-loop system maintains recursive feasibility even in the presence of sudden changes in the target reference.