Most determinations of elastic constants of anisotropic solids are based on acoustic wave phase velocity measurements [1,2]. The methods for recovering elastic constants from phase velocity data are fairly straightforward and well-established. For high symmetry directions certain elastic constants, or combinations thereóf, can be obtained directly from individual velocity measurements [3]. When data pertaining to off-symmetry directions is used, usually an optimized fitting procedure is invoked to obtain the full set of elastic constants [4]. There is growing use of techniques that measure the group velocities of acoustic waves or phonons in solids. In an anisotropic solid the phase and group velocities do not in general coincide, even in the absence of dispersion and attenuation. Existing methods for recovering elastic constants from phase velocity data are therefore not directly applicable. The aim of this paper is to present a novel method that allows the elastic constants of an anisotropic solid to be recovered from group velocities measured in arbitrary directions in a sample. We demonstrate the method by applying it to computer generated velocity data on cubic and transversely isotropic solids and to experimental data obtained on single crystals of silicon.