Elastic properties of thin supported films can be derived from the dispersion relations of surface acoustic waves (SAWs), which depend on the properties of the films themselves. Among the techniques for the measurement of SAW velocities surface Brillouin scattering (SBS) of visible light probes SAWs at the shorter wavelengths (around 0.5 micrometers ), allowing resolution down to nanometric films. Since SAW velocities can be computed as function of elastic constants and mass density of both the film and the substrate, of film thickness and of wavevector, the elastic properties can be obtained by fitting the computed velocities of the measured ones. Namely, if film thickness and density are independently measured, e.g. by X-ray reflectivity and X-ray diffraction, the elastic constants of the film can be derived by a Generalized Least Squares estimator, with corresponding confidence intervals. Accurate derivation of elastic constants requires highly accurate SAW velocity measurements. Some examples are considered in detail: diamond-like carbon films on silicon substrate and titanium silicide films, showing that elastic constants of thin films can be determined by SBS measurements with precisions ranging from reasonable to very good.