This paper is concerned with periodic dynamic event-triggered H_∞ control for networked switched linear systems with both mode-dependent average dwell time (MDADT) and asynchronous switching. Compared with the existing event-triggered results of the switched systems, the periodic dynamic event-triggered scheme introduces dynamic event-triggered parameters and has the minimum lower bound of event-triggered interval, which can effectively reduce the number of data packages and avoid the Zeno behavior. By using the MDADT approach and the multiple Lyapunov functions technique, sufficient conditions are proposed so that the close-loop system is exponential stable with a weighted H_∞ performance. Furthermore, the corresponding criterion is proposed to design a set of controllers for the switched system with asynchronism. Finally, an example is provided to verify the superiority of the acquired results.