An electronic device may be provided with electronic components such as radio-frequency transceiver integrated circuits and other integrated circuits that are be sensitive to electromagnetic interference. Metal structures are configured to serve both as heat sinking structures for the electrical components and electromagnetic interference shielding. Components are mounted to the substrate using solder. Metal fence structures are also soldered to the substrate. Each metal fence has an opening that covers a respective one of the components. A thermally conductive elastomeric gap filler pad is mounted in the opening. A metal heat spreading structure is electrically shorted to the fence using a conductive gasket that surrounds the gap filler pad so that the structure serves as an electromagnetic interference shield. Heat from the component travels through the gap filler pad to the metal heat spreading structure so that the heat spreading structure may laterally spread and dissipate the heat.