State-of-the-art optical switches for coupling pulses into and/or out of resonators are based on either the electro-optic or the acousto-optic effect in transmissive elements. In high-power applications, the damage threshold and other nonlinear and thermal effects in these elements impede further improvements in pulse energy, duration, and average power. We propose a new optomechanical switching concept which is based solely on reflective elements and is suitable for switching times down to the ten-nanosecond range. To this end, an isolated section of a beam path is moved in a system comprising mirrors rotating at a high angular velocity and stationary imaging mirrors, without affecting the propagation of the beam thereafter. We discuss three variants of the concept and exemplify practical parameters for its application in regenerative amplifiers and stack-and-dump enhancement cavities. We find that optomechanical pulse picking has the potential to achieve switching rates of up to a few tens of kilohertz while supporting pulse energies of up to several joules.