The thermal properties of a material with a spatiotemporal modulation, in the form of a traveling wave, in both the thermal conductivity and the specific heat capacity are studied. It is found that these materials behave as materials with an internal convectionlike term that provides them with nonreciprocal properties, in the sense that the heat flux has different properties when it propagates in the same direction or in the opposite one to the modulation of the parameters. An effective medium description is presented which accurately describes the modulated material, and numerical simulations support this description and verify the nonreciprocal properties of the material. It is found that these materials are promising candidates for the design of thermal diodes and other advanced devices for the control of the heat flow at all scales.