This paper proposes the use of a randomized kinodynamic planning technique to synthesize dynamic motions for cable-suspended parallel robots. Given two mechanical states of the robot, both with a prescribed position and velocity, the method attempts to connect them by a collision-free trajectory that respects the joint and force limits of the actuators, keeps the cables in tension, and takes the robot dynamics into account. The method is based on the construction of a bidirectional rapidly-exploring random tree over the state space. Remarkably, the technique can be used to cross forward singularities of the robot in a predictable manner, which extends the motion capabilities beyond those demonstrated in previous work. The paper describes experiments that show the performance of the method in point-topoint operations with specific cable-driven robots, but the overall strategy remains applicable to other mechanism designs.