This article proposes a resource-aware triggering-based dynamic controller codesign scheme for Markov jump systems (MJSs) to fulfill both disturbance rejection and computational resources saving goals. Specifically, a self-trigger strategy is presented for precomputing the next execution time for state measurement, executing the controller, and updating the actuators using the current sampled information. To achieve a desired system performance and reduce the resource occupation, the self-triggering parameters and the gains of state feedback controller are codesigned. Moreover, a dynamic triggering technique is employed to improve the system performance by adapting the trigger coefficients to the different subsystems of MJSs. The effectiveness of the proposed method is demonstrated via two examples.