We employed an easy and direct method to measure the thermal conductivity of epoxy in the liquid (nanofluid) and solid (nanocomposite) states using both rodlike and platelet-like carbon-based nanostructures. Comparing the experimental results with the theoretical model, an anomalous enhancement was obtained with multiwall carbon nanotubes, probably due to their layered structure and lowest surface resistance. Puzzling results for functionalized graphene sheet nanocomposites suggest that phonon coupling of the vibrational modes of the graphene and of the polymeric matrix plays a dominant role on the thermal conductivities of the liquid and solid states.PACS: 74.25.fc; 81.05.Qk; 81.07.Pr.