Thermal radiation heat transport within prolate and oblate ellipsoidal cavities was examined. The axisymmetric anisotropy of the cavity shape gives rise to a thermal radiation conductivity tensor with principal axes components parallel (λr, ‖) and perpendicular (λr, ⊥) to the symmetry axis. The prolate (λr, ⊥) and oblate (λr, ⊥) transverse components are calculated and compared with well-known results from the kinetic theory of transport across cylindrical and within slit void geometries. The use of λr, ⊥ and λr, ⊥, along with earlier results for λr, ‖ and λr, ‖, in well-known effective conductivity equations for spheroidal inclusions within a solid matrix, provides a means to rigorously treat cavity orientation and shape in high-temperature heat transport across porous materials. To facilitate the calculations and produce readily usable equations, a variational principle is used.