The thermal conductivity of polymer composites is measured for several tubular carbon nanofillers (nanotubes, fibres, and whiskers). The highest enhancement in the thermal conductivity is observed for functionalized multiwalled carbon nanotubes (90% enhancement for 1 vol. %) and Pyrograf carbon fibres (80%). We model the experimental data using an effective thermal medium theory and determine the thermal interface resistance (RK ) at the filler-matrix interface. Our results show that the geometry of the nanofibres and the interface resistance are two key factors in engineering heat transport in a composite.