We propose a motion planner for cable-driven payload transportation using multiple unmanned aerial vehicles (UAVs) in an environment cluttered with obstacles. Our planner is kinodynamic, i.e., it considers the full dynamics model of the transporting system including actuation constraints. Due to the high dimensionality of the planning problem, we use a hierarchical approach where we first solve the geometric motion planning using a sampling-based method with a novel sampler, followed by constrained trajectory optimization that considers the full dynamics of the system. Both planning stages consider inter-robot and robot/obstacle collisions. We demonstrate in a software-in-the-loop simulation and real flight experiments that there is a significant benefit in kinodynamic motion planning for such payload transport systems with respect to payload tracking error and energy consumption compared to the standard methods of planning for the payload alone. Notably, we observe a significantly higher success rate in scenarios where the team formation changes are needed to move through tight spaces.