The existence of a net heat current of conductive thermal waves is demonstrated even in the absence of a mean temperature gradient. This effect, which we called heat shuttling, is generated by the temperature-dependent thermal conductivity of materials excited with a thermal excitation periodically modulated in time. We show that this modulation gives rise to a heat current superimposed on the one generated by the mean temperature gradient, which enhances the heat transport when the thermal conductivity increases with temperature. By contrast, if the thermal conductivity decreases as temperature increases, the thermal-wave heat current inverts its direction and reduces the total heat flux. The reported shutting effect is sensitive to the amplitude of the periodic thermal excitation, which can facilitate its observation and application to harvest energy from the temperature variations of the environment.