A structure for cooling an electronic device is disclosed. The structure includes a compressible top layer disposed over the electronic device. The structure further includes a plurality of rigid elements disposed between the top layer and the electronic device for providing a heat path from the electronic device and wherein the plurality of rigid elements provide mechanical compliance. In another alternative, the structure further includes a conformable heat-conducting layer disposed over the electronic device, wherein a bottom end of the plurality of rigid elements is coupled to the conformable heat-conducting layer.