Seamless advancements in the electronics industry lead to high heat fluxes from very limited thermal real estates. Use of natural convection air cooling is of interest to meet some of the low flux cooling needs, while active cooling techniques via liquid or forced convection are the methods of choice. In natural convection heat transfer applications, the components used for cooling may represent a significant portion of the overall weight of the system. Consequently, advanced materials are of interest in such applications, as they may substantially reduce the total size and weight of the system. Many of these advanced materials have anisotropic thermophysical properties, hence the control of thermal conductivity is crucial. This paper is motivated to address the lack of understanding of the use of anisotropic advanced materials in natural convection environments. Numerical simulations are carried out to test the performance of heat sinks made of such materials and comparisons are made with the heat sinks of traditional engineering materials under the same conditions. The results demonstrate that the total weight of the system may be reduced drastically with the use of advanced materials relative to the most commonly used heat sink materials at the same thermal performance. A figure of merit (FOM) is proposed to compare the thermal performance of different heat sinks. Total resistance, conduction and convection resistances, and performance-related FOM values for each heat sink are presented. It is shown that the conduction thermal resistance is dominant at lower fin thicknesses for sparse heat sinks while it is negligible for dense heat sinks. Pyrolytic graphite-based heat sinks demonstrate the best thermal performance, while carbon-foam heat sinks produce the highest FOM values due to the material's low density.