As motivated by the recent discovery of heat pulses propagating in dielectric crystals at low temperature, a continuum theory of thermoelasticity, which is modified to include the effect of thermal phonon relaxation, is applied to investigate the propagation of plane harmonic waves in unbounded anisotropic solids. Four characteristic wave speeds are found, three being analogous to those of isothermal or adiabatic elastic waves. The fourth wave, which is predominantly a temperature disturbance, corresponds to the heat pulses known also as the second sound. Velocity, slowness, and wave surfaces of the thermoelastic waves are analyzed and are illustrated with numerical and graphical results for NaF and solid helium crystals. A new definition of the group velocity for waves in a dissipative and dispersive anisotropic medium is proposed and is calculated and compared with the energy transport speed of thermoelastic waves.