We demonstrate that the time-averaged far-field thermal radiation between macroscopic bodies can be different from Planck's limit when their temperature difference is periodically modulated. This difference arises from the nonlinear dependence of the radiative heat flux on temperature and persists even for temperature-independent emissivities. By contrast, when the material emissivity depends on temperature, the modulated thermal radiation is driven by the second temperature derivative of emissivity and increases with the square of the temperature oscillations' amplitude. These temperature fluctuations generate thermal radiation even without a mean temperature difference. Our findings thus reveal a novel mechanism to control and modulate far-field radiation with potential applications in thermal management and energy harvesting.