In this paper, the problems of stochastic stability and sliding mode control of a class of continuous-time Markovian jump systems with partly unknown transition probabilities are considered. The proposed approach is quite general and covers both cases in which transition probability rates are completely known and completely unknown. By the assistance of the free-connection weighting matrices, sufficient conditions guaranteeing the existence of linear switching surface and the stochastic stability of sliding mode dynamics are obtained in terms of linear matrix inequalities (LMIs). Then, a sliding mode controller is designed such that the state trajectories of the closed-loop system reach the desired sliding surface in a finite time and maintain there for all subsequent times. Both the sliding surface and sliding mode control law are obtained by means of LMI feasibility problems. Finally, a numerical example is given to show the potentials and effectiveness of the proposed method.