We demonstrate that the energy transport of surface phonon-polaritons can be large enough to be observable in a crystal made up of a three-dimensional assembly of nanorods of silicon carbide. The ultralow phonon thermal conductivity of this nanostructure along with its high surface area-to-volume ratio allows the predominance of the polariton energy over that generated by phonons. The dispersion relation, propagation length, and thermal conductance of polaritons are numerically determined as functions of the radius and temperature of the nanorods. It is shown that the thermal conductance of a crystal with nanorods at 500 K and diameter (length) of 200 nm (20 μm) is 0.55 nW·K-1, which is comparable to the quantum of thermal conductance of polar nanowires.