This study scrutinizes the extended dissipative problem for Markovian jump generalized neural networks with asynchronous mode-dependent time-varying interval delayed states. A suitable Lyapunov–Krasovskii functional and a new bounding technique can derive delay-dependent results to achieve an extended dissipative performance index. Jensen’s inequality, reciprocally convex combination, and a novel integral inequality technique are utilized in this paper. The proposed criteria are reliable since many components are included in the unified neural network model, Markovian jumping, and time-varying delay with asynchronous modes. In this work, the systemic and time-varying delay modes are expressed asynchronously, which means that they depend on different jumping modes. Four numerical examples show the effectiveness and usefulness of the presented results.