In an integrated circuit packaging structure, such as in an MCM or in a SCM, a compliant thermally conductive material is applied between a heat-generating integrated circuit chip and a substrate attached thereto. Raised regions are defined on the back side of the chip aligned to areas of a higher than average power density on the front active surface of the chip such that a thinner layer of the compliant thermally conductive material is disposed between the chip and the substrate in this area after assembly thereof resulting in a reduced “hot-spot” temperature on the chip. In an exemplary embodiment, the substrate includes one of a heat sink, cooling plate, thermal spreader, heat pipe, thermal hat, package lid, or other cooling member.