An electronic device may have a housing. Components such as keys may be mounted in openings in the housing. The housing may include a display housing portion and a base portion that are coupled using hinge structures. The keys may be used to form a keyboard in the base portion of the housing. Sensitive circuitry such as sensitive wireless circuitry may be shielded from electromagnetic interference by shielding the openings with a layer of conductive material. The layer of conductive material may be a metal layer that forms part of a reflector for a backlight structure. The backlight structure may be used to provide backlight for keys in the keyboard. The keyboard may include a support structure having openings through which the backlight passes to reach the keys. The openings may be bridged using conductive paths so that the support structure serves as a radio-frequency shielding structure.