Diffusing-wave spectroscopy was used to measure the microscopic dynamics of grains in the interior of a three-dimensional flow of sand. The correlation functions show that minutely separated grains fly from collision to collision with large random velocities. On a time scale 10(3) to 10(4) times longer than the average time between collisions, the grains displayed slow, collective rearrangements, which, at the long-time limit, produced diffusive dynamics.