This paper investigates the problem of mixed event/time-triggered H_∞ control of Markov jump systems suffered by unreliable communication channels and limited bandwidth constraints. To deal with the finite bandwidth constraints, a distributed mixed event/time-triggered mechanism is firstly designed based on triggered errors, auxiliary variables and switching times to determine the triggered instants of each sensor node. Secondly, a novel TOD (try-once-discard) scheduling protocol is proposed, which ensures that only a unique sensor node is connected to the network at the triggered instants. Furthermore, the data packet loss caused by unreliable communication channels is modeled as Bernoulli distribution. The random sequence and scheduling matrix sequence are constructed based on the data transmitted at the triggered times, and the data reception is modeled. Based on this, by using linear matrix inequality, the sufficient conditions are obtained to ensure that the closed-loop Markov jump system is stochastic stable without disturbances and finite gain $L_2$ stochastic stable with disturbances, and the design methods of feedback gain matrix are given. Finally, the effectiveness of the proposed method is verified through simulation.