We report ultrafast transient-grating measurements of crystals of the three-dimensional Dirac semimetal cadmium arsenide, Cd₃As₂, at both room temperature and 80 K. After photoexcitation with 1.5-eV photons, charge-carriers relax by two processes, one of duration 500 fs and the other of duration 3.1 ps. By measuring the complex phase of the change in reflectance, we determine that the faster signal corresponds to a decrease in absorption, and the slower signal to a decrease in the light's phase velocity, at the probe energy. We attribute these signals to electrons' filling of phase space, first near the photon energy and later at lower energy. We attribute their decay to cooling by rapid emission of optical phonons, then slower emission of acoustic phonons. We also present evidence that both the electrons and the lattice are strongly heated.