In terms of elastic anisotropy, many rocks may be considered to have orthorhombic symmetry. Experimentally determining the nine independent elastic coefficients required for this case remains challenging. Elastic coefficients are most often found from measurements of the phase velocity in a variety of directions throughout a material, but finding this plane-wave velocity is problematic. Here, quasi-P and quasi-S phase speeds are found using the τ–p transformation through a composite material of orthorhombic symmetry. Arrays of specially constructed transducers (0.65 MHz) with different modes of vibration were placed on a rectangular prism of the material. More than 620 individual measures of phase speed were obtained at different directions and subsequently used in a generalized least-squares inversion that yields the required elastic coefficients. The analysis does not account for the effects of wave-speed dispersion evident in the waveforms acquired in the composite material. This dispersion is particularly severe for the in-plane, quasi-S polarization and is possibly a consequence of the fine layered structure of the material.