We consider the problem of controlling multiple robots manipulating and transporting a payload in three dimensions via cables. Individual robot control laws and motion plans enable the control of the payload (position and orientation) along a desired trajectory. We address the fact that robot configurations may admit multiple payload equilibrium solutions by developing constraints for the robot configuration that guarantee the existence of a unique payload pose. Further, we formulate individual robot control laws that enforce these constraints and enable the design of non-trivial payload motion plans. Finally, we propose two quality measures for motion plan design that minimize individual robot motion and maximize payload stability along the trajectory. The methods proposed in the work are evaluated on a team of aerial robots in experimentation.