Highly anharmonic bonding, quantified by the Grüneisen parameter, can lead to low glass-like lattice thermal conductivity and is therefore desirable in thermoelectric materials. However, most investigations of anharmonic effects on lattice thermal conductivity emphasize phonon-phonon scattering, while overlooking the impact of anharmonicity on bond strength and sound velocity at elevated temperatures. The elastic moduli of most thermoelectric materials decrease gradually with increasing temperature due to thermal expansion (increased bond length leads to weaker bonds). 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, and are rarely accounted for when modeling thermal conductivity. In this work, we combine high-temperature resonant ultrasound spectroscopy and in-situ X-ray diffraction to characterize the temperature-dependent elastic constants and anisotropic thermal expansion in several classes of thermoelectric materials (e.g., Zintl antimonides, Cu2ABTe4 stannites, GexSb2Te3+x alloys, and others). We have observed materials with high Grüneisen parameters and rapid softening at high temperature, as well as cases in which the lattice actually stiffens despite increasing average bond length. At both extremes, the temperature-dependence of the bond strength is found to have a significant impact on the speed of sound, lattice thermal conductivity and ultimately the performance of these materials.