We compute the force on a small neutral polarizable object moving at velocity v--> relative to a photon gas equilibrated at a temperature T. We find a drag force linear in v-->. Its physical basis is related to that in recent formulations of the dissipative component of the Casimir force, i.e., the change in photon momentum in emission and absorption between the moving body and the stationary thermal bath. We estimate the strength of this universal drag force for different dielectric response functions and comment on its relevance in various contexts, especially to radiation-matter coupling in the cosmos.