Electronic devices such as computers and handheld devices are provided. The electronic devices may have electrical components such as touch pads and displays. The displays may be touch screen displays. The touch pads and touch screen displays may be formed from touch panels having touch sensors mounted to a planar dielectric member. The displays may be formed from light-emitting structures mounted to a planar dielectric member. The planar dielectric members in the touch panels and displays may have one or more antenna traces that form antennas for the electronic devices. Electrical connectors such as spring-loaded pins, springs, and flexible transmission line structures may be used to form radio-frequency signal paths between the antenna traces on a planar dielectric member and radio-frequency transceiver integrated circuits mounted on a circuit board in an electronic device. The electronic device may have conductive housing walls to which the planar dielectric member is mounted.