Recent studies of SWNT/polymer nanocomposites identify the large interfacial thermal resistance at nanotube/nanotube junctions as a primary cause for the only modest increases in thermal conductivity relative to the polymer matrix. To reduce this interfacial thermal resistance, we prepared a freestanding nanotube framework by removing the polymer matrix from a 1 wt % SWNT/PMMA composite by nitrogen gasification and then infiltrated it with epoxy resin and cured. The SWNT/epoxy composite made by this infiltration method has a micron-scale, bicontinuous morphology and much improved thermal conductivity (220% relative to epoxy) due to the more effective heat transfer within the nanotube-rich phase. By applying a linear mixing rule to the bicontinuous composite, we conclude that even at high loadings the nanotube framework more effectively transports phonons than well-dispersed SWNT bundles. Contrary to the widely accepted approaches, these findings suggest that better thermal and electrical conductivities can be accomplished via heterogeneous distributions of SWNT in polymer matrices.