Performing highly agile dynamic motions, such as jumping or running on uneven stepping stones has remained a challenging problem in legged robot locomotion. This paper presents a framework that combines trajectory optimization and model predictive control to perform robust and consecutive jumping on stepping stones. In our approach, we first utilize trajectory optimization based on full-nonlinear dynamics of the robot to generate periodic jumping trajectories for various jumping distances. A jumping controller based on a model predictive control is then designed for realizing smooth jumping transitions, enabling the robot to achieve continuous jumps on stepping stones. Thanks to the incorporation of MPC as a real-time feedback controller, the proposed framework is also validated to be robust to uneven platforms with unknown height perturbations and model uncertainty on the robot dynamics.