This note investigates the stabilization problem for a class of linear uncertain networked control systems with random communication time delays. Both sensor-to-controller and controller-to-actuator random-network-induced delays are considered. Markov processes are used to model these random-network-induced delays. Based on the Lyapunov-Razumikhin method a mode-dependent state feedback controller is proposed to stabilize this class of systems. The existence of such a controller is given in terms of the solvability of bilinear matrix inequalities, which are to be solved by a newly proposed algorithm. A numerical example is used to illustrate the validity of the design methodology.