The thermal transport properties of random network, single-walled carbon nanotube (SWNT) films were assessed using Raman spectroscopy. Two types of SWNT films were investigated: single-layer and stacked. The single-layer films were fabricated by aerosol chemical vapour deposition and subsequent direct dry deposition, while the stacked films were prepared by placing the single-layer films on top of one another. The anisotropy of the network structures of each of these films was evaluated based on the angular dependence of the optical absorbance spectra. The results show that the anisotropy of the films decreases with increasing film thickness in the case of the single-layer films, and that the film anisotropy is preserved during the stacking process. The sheet thermal conductance is proportional to the SWNT area density in the case of stacked films, but is reduced with increasing thickness in the case of single-layer films. This effect is explained by a change in the network morphology from a two-dimensional anisotropic structure to the more isotropic structure. This work demonstrates the fabrication of low-density films with high sheet thermal conductance through the stacking of thin SWNT films.