The thermal response of particles subjected to time-dependent temperature perturbations in the surrounding medium, including radiation heat transfer, is studied. The continuous medium containing the dispersion of particles is assumed to be weakly participating, having a small but non-zero absorption coefficient. The particles in suspension are either in Stokes flow or stationary, so that the mechanisms of heat transfer between the particles and the continuous phase are of diffusive and radiative nature only. The general solution for the temperature response of the particles to time-dependent perturbations in the continuous phase is derived. The method used to derive the general solution consists of including the radiative effects in an integro-differential equation that describes the temperature history of the particles. A fractional-differential operator is then applied to the radiation-diffusion equation. The resulting equation is solved exactly by the method of variation of parameters for the temperature potential between the particles and the medium. Linear and harmonic perturbations are analyzed and discussed.