Two images of Cassiopeia A obtained at 24 micrometers with the Spitzer Space Telescope over a 1-year time interval show moving structures outside the shell of the supernova remnant to a distance of more than 20 arc minutes. Individual features exhibit apparent motions of 10 to 20 arc seconds per year, independently confirmed by near-infrared observations. The observed tangential velocities are at roughly the speed of light. It is likely that the moving structures are infrared echoes, in which interstellar dust is heated by the explosion and by flares from the compact object near the center of the remnant.