Cross-flow turbines, also known as vertical-axis turbines, are attractive for power generation from wind and water currents. Some cross-flow turbine designs optimize unsteady fluid forces and maximize power output by controlling blade kinematics within one rotation. One established method is to dynamically pitch the blades. Here we introduce a mechanically simpler alternative: optimize the turbine rotation rate as a function of angular blade position. We demonstrate experimentally that this approach results in a 59% increase in power output over standard control methods. Analysis of fluid forcing and blade kinematics suggest that power increase is achieved through modification of the local flow conditions and alignment of fluid force and rotation rate extrema. The result is a low-speed, structurally robust turbine that achieves high efficiency and could enable a new generation of environmentally benign turbines for renewable power generation.