This paper investigates the stability problem of Markovian neural networks (MNNs) with time delay. First, to reflect more realistic behaviors, more generalized transition rates are considered for MNNs, where all transition rates of some jumping modes are completely unknown. Second, a new approach, namely time-delay-dependent-matrix (TDDM) approach, is proposed for the first time. The TDDM approach is associated with both time delay and its time derivative. Thus, the TDDM approach can fully capture the information of time delay and would play a key role in deriving less conservative results. Third, based on the TDDM approach and applying Wirtinger's inequality and improved reciprocally convex inequality, stability criteria are derived. In comparison with some existing results, our results are not only less conservative but also involve lower calculation complexity. Finally, numerical examples are provided to show the effectiveness and advantages of the proposed results.