In thermoelectric materials, strongly anharmonic bonding is desired due to the resulting phonon-phonon scattering and reduced lattice thermal conductivity. Typically, an average Grüneisen parameter obtained from thermal expansion measurements, is used to quantify the degree of anharmonicity in the solid. Although a correlation between the Grüneisen parameter and the slope of the elastic moduli versus temperature has long been recognized, this relationship and its consequences have not been systematically investigated. In this work, we employ high-temperature resonant ultrasound spectroscopy to characterize the temperature-dependent sound velocity and elastic constants of diverse thermoelectric materials, (e.g., Zintl antimonides, Cu2ABTe4 stannites, and other chalcogenides). These results are combined with in-situ X-ray diffraction to obtain thermal expansion coefficients, revealing a greater degree of bonding anharmonicity along “softer” crystallographic directions. We compare several published approaches to calculating the average Grüneisen parameter, to identify the most relevant indicator of the resulting phonon-phonon scattering rate.