Different types of graphene-related materials (GRM) are industrially available and have been exploited for thermal conductivity enhancement in polymers. These include materials with very different features, in terms of thickness, lateral size and composition, especially concerning the oxygen to carbon ratio and the possible presence of surface functionalization. Due to the variability of GRM properties, the differences in polymer nanocomposites preparation methods and the microstructures obtained, a large scatter of thermal conductivity performance is found in literature. However, detailed correlations between GRM-based nanocomposites features, including nanoplatelets thickness and size, defectiveness, composition and dispersion, with their thermal conductivity remain mostly undefined. In the present paper, the thermal conductivity of GRM-based polymer nanocomposites, prepared by melt polymerization of cyclic polybutylene terephtalate oligomers and exploiting 13 different GRM grades, was investigated. The selected GRM, covering a wide range of specific surface area, size and defectiveness, secure a sound basis for the understanding of the effect of GRM properties on the thermal conductivity of their relevant polymer nanocomposites. Indeed, the obtained thermal conductivity appeares to depend on the interplay between the above GRM feature. In particular, the combination of low GRM defectiveness and high filler percolation density was found to maximize the thermal conductivity of nanocomposites.