Abstract <p>This manuscript investigates harmonic plane wave propagation in a time differential Moore–Gibson–Thompson thermoelastic medium. It is noted that six possible plane harmonic waves may propagate at different speeds. Among these, two are transverse waves, while the other four are coupled longitudinal waves. The transverse waves are decoupled, undamped over time, and propagate independently at a speed unaffected by the thermal field. The four longitudinal plane waves exhibit coupling, temporal damping, and dispersion due to the thermal influence of the medium. A&#xa0;longitudinally quasi-elastic wave decays exponentially over time, with its amplitude diminishing to zero as time progresses toward infinity. A stationary quasi-thermal wave also decays exponentially to zero over time. Additionally, there are two possible dilatational quasi-thermal propagating waves with varying rates of time damping, or there could be a single time-harmonic dilatational thermal wave, depending on the time delay value. The problem of surface waves is also discussed for Moore–Gibson–Thompson thermoelasticity. The surface of the half-space is assumed to be traction-free and able to exchange heat freely with the surrounding medium. The dispersion relation for the surface wave is explicitly formulated, and the secular equation is derived. Numerical simulations are carried out for both plane and surface waves within a specified model. The computed results are visually depicted, and a summary analysis of these outcomes is provided.</p>