Precise trajectory tracking and stiffness modulation for tensegrity mechanisms are a challenging topic that can open new horizon of applications for this type of systems. This paper presents a new control strategy of tensegrity mechanisms using a model predictive controller (MPC). Based on a dynamic model, the proposed approach allows to track trajectories with low and relatively high dynamics as well as to modulate the mechanism stiffness by changing only the controller's parameters. Trajectories of 30s, 5s and 1s are performed showing a trajectory tracking improvement of up to 64% in the root mean square error when compared to literature results.