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 (lambda(r), (parallel to)) and perpendicular (lambda(r, perpendicular to)) to the symmetry axis. The prolate (lambda(r, perpendicular to)) and oblate (<(lambda)over tilde>(r, perpendicular to)) 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 lambda(r, perpendicular to) and <(lambda)over tilde>(r, perpendicular to), along with earlier results for lambda(r, parallel to) and <(lambda)over tilde>(r, parallel to), 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.