Thermal analysis of planar and near-square semiconductor device chips employing angular Fourier-series (AFS) expansion is presented for the first time. The determination of the device peak temperature using AFS requires only a single two-dimensional computation, while full three-dimensional temperature distribution can be obtained, if desired, by successively adding higher-order Fourier terms, each of which requires a separate 2D computation. The AFS method is used to compare the heat spreading characteristics of AlGaN/GaN high-electron-mobility transistors (HEMTs) fabricated on silicon, silicon carbide, and synthetic diamond. We show that AlGaN/GaN HEMTs built using GaN/diamond technology can offer better than half the thermal resistance of GaN/SiC HEMTs under worst-case cooling conditions. Furthermore, we show that, if left unmanaged, an inherent and non-negligible thermal boundary resistance due to the integration of semiconductor epilayers with non-native substrates will dampen the benefits of highly conductive substrates such as SiC and diamond.