A variational principle is formulated for radiant heat transfer through the void spaces and conduction in the solid, of an arbitrary void-solid system with diffusely reflecting surfaces. Variational upper bounds on the effective conductivity of a void-solid suspension are expressed in terms of certain averages characterizing the random geometry. A radiation conductivity upper bound, calculated for a model porous medium generated by randomly placed, overlapping spheres, is compared with well-known results from kinetic theory and radiant heat transport. The variational principle provides a means to obtain useful estimates and upper bounds on the thermal radiation conductivity, that include multiple and anisotropic diffusive scattering in a bed of large particles.