An unstructured finite volume scheme is applied to the solution of sub-micron heat conduction problems. The phonon Boltzmann transport equation (BTE) in the relaxation time approximation is considered. The similarity between the radiative transfer equation (RTE) and the BTE is exploited in developing a finite volume scheme for the BTE. The spatial domain is divided into arbitrary unstructured polyhedra, the angular domain into control angles and the frequency domain into frequency bands and conservation equations for phonon energy are written. The unsteady wave propagation term; not usually present in thermal radiation problems, is differenced using a fully implicit scheme. A sequential multigrid scheme is applied to solve the nominally linear set. Isotropic scattering due to a variety of mechanisms such as impurity and Umklapp scattering is considered. The numerical scheme is applied to a variety of sub-micron conduction problems, both unsteady and steady. Favorable comparison is found with the published literature and with exact solutions.