Temperature-rate-dependent thermoelasticity is a theory of thermoelasticity in which two relaxation times are introduced into the equations of classical thermoelasticity. An important consequence of this theory is that heat now travels at a finite speed rather than the infinite speed implied by the diffusion equation. In an anisotropic temperature-rate-dependent thermoelastic material, it is found that four plane harmonic waves may propagate in any direction, all dispersive and attenuated, yet all are stable in the sense that their amplitudes remain bounded. An alternative theory that forces heat to travel at finite speed is generalized thermoelasticity in which the rate of change of heat flux also appears in the heat conduction equation, thereby introducing a relaxation time. Two different methods of combining the effects of temperature-rate-dependent thermoelasticity and generalized thermoelasticity are discussed, and it is found that at least some of the four waves become unstable.