Using high-resolution microwave sky maps made by the Atacama Cosmology Telescope, we for the first time present strong evidence for motions of galaxy clusters and groups via microwave background temperature distortions due to the kinematic Sunyaev-Zel'dovich effect. Galaxy clusters are identified by their constituent luminous galaxies observed by the Baryon Oscillation Spectroscopic Survey, part of the Sloan Digital Sky Survey III. We measure the mean pairwise momentum of clusters, with a probability of the signal being due to random errors of 0.002, and the signal is consistent with the growth of cosmic structure in the standard model of cosmology.