The CRESST dark matter search employs phonon-mediated detectors with CaWO4 absorbers, prompting renewed interest in its thermal properties. We use phonon-imaging techniques to measure temporal and spatial phonon flux distributions in a [001]-oriented CaWO4 crystal. Data from the crystal and simulations based on continuum elasticity theory are used to calculate values of the elastic constants. The measured elastic constant values are consistent with those measured previously by Farley et al. via ultrasonic pulses, except for C13 and C16, which differ by 15% and 8%, respectively.