A sampled-data delayed feedback control framework is proposed for stabilizing both periodic solutions of linear periodic systems and unstable periodic orbits of nonlinear systems. The proposed control framework utilizes the difference between two consecutive samples of the state as a feedback input. The controller is turned on and off periodically in an alternating fashion, allowing us to obtain a monodromy matrix for the closed-loop system under our proposed control law. We investigate asymptotic behavior of the closed-loop system state trajectories through an analysis of the obtained monodromy matrix. Finally, we present a numerical example to demonstrate the efficacy of our approach.