A sampled-data Pyragas-type delayed feedback control framework is proposed for stabilizing both periodic solutions of linear periodic systems and unstable periodic orbits of nonlinear systems. The proposed framework uses the difference between two consecutive samples of the state as a feedback input. Furthermore, the controller is turned on and off periodically in an alternating fashion, allowing one to obtain a monodromy matrix for the closed-loop system. Asymptotic behavior of the closed-loop system state trajectories is investigated through an analysis of the obtained monodromy matrix. Then, an efficient feedback gain design method is developed by exploiting the properties of the monodromy matrix. Finally, the efficacy of the proposed framework is demonstrated with a numerical example.