A carbon nanotube (CNT) array is patterned on a substrate. The substrate can be a microelectronic die, an interposer-type structure for a flip-chip, a mounting substrate, or a board. The CNT array is patterned by using a patterned metallic seed layer on the substrate to form the CNT array by chemical vapor deposition. The patterned CNT array can also be patterned by using a patterned mask on the substrate to form the CNT array by growing. A computing system that uses the CNT array for heat transfer from the die is also used.