An innovative cooling tube assembly is disclosed for use in cooling the liner of a combustor in a gas turbine engine. Each cooling tube has a serpentine shape that conforms to the contour of the liner as the tube extends from one end of the combustor to the other to conduct cooling air in a counterflow direction to the exhaust gases. The cooling air gains heat, thereby cooling the liner wall, and is delivered in the proximity of the burners where its heat aids in combustion. An assembly of cooling tubes is provided for mounting on either the cool side or the hot side of the combustor liner, the assembly comprising a plurality of nested cooling tubes. The cooling tubes may be fabricated of a metallic material and either brazed to the liner or supported by pins inserted through the liner. The cooling tubes may also be fabricated of a CMC material by an innovative method in which an expendable mandrel of the desired cooling tube shape is inserted into one or more sleeves of a woven tubular fabric bound to a fabric shell formed in the shape of the combustor liner and densified in a furnace.