Nanomultilayers are complex architectures of materials stacked in sequence with layer thicknesses in the nanometer range. Their application in microelectronics is challenged by their thermal stability, conductivity and interface reactivity which can compromise their performance and usability. By using different materials as thermal barriers and by changing their thickness, it is possible to manipulate interfacial effects on thermal transport. In this work, we report on the thermal conductivity of Cu/W, Cu/Ta, and Cu/TaN sputter deposited nanomultilayers with different thicknesses. The resistive interfacial effects are rationalized and discussed also in relation to the structural transformation into a nano-composite upon high temperature annealing.