Thermal rectification based on a new strategy of nanostructuration has been assessed with nonequilibrium molecular dynamics and wave-packet propagation simulations. The use of asymmetric crystallinecore/amorphous-conical-shell nanowires creates a direction-dependent thermal conductivity due to variable axial and radial phonon propagation/confinement. The origin of this effect is related to the combination of a crystalline/amorphous interface parallel to the heat flux, together with the variable amount of amorphous coating due to the conical shell of the nanowire. Physical insights of the rectification are given by the mean free path of phonons, the axial and radial energy transfer, the energy diffusivity, and the vibrational density of states restricted to different constitutive elements of the nanowire.