A weak thermal conductivity (TC) of a polymer can be modified by inclusion of nanoparticles with high TC. Here we study the TC enhancement in epoxy resin (ER) based composites by incorporation of carbon nanotubes (CNTs) and demonstrate that the enhancement depends critically on the alignment of CNTs. The highest effect in TC enhancement (18.9) was obtained in ER with vertically aligned multiwall CNTs (VANTs) and in ER with horizontally aligned nanotubes (HANTs) (6.5). We analyze the influence of intrinsic structural factors of CNTs as well as extrinsic factors limiting the enhancement of the composite TC. The dynamics of heat propagation in ER/VANT, a strongly anisotropic and heterogeneous system, was studied experimentally, using laser flash apparatus (LFA), and by computer simulation, applying a coaxial cylinder model. It was found that the thermal resistivity CNT-ER interface to be a key extrinsic factor limiting the dynamics of the heat propagation. We show that these dynamics and the interface resistivity can be efficiently studied using the LFA technique.