An electronic device may have a housing in which components are mounted that produce heat. The heat producing components may be light-emitting diodes mounted on a flexible printed circuit in a display backlight, may be integrated circuits, or may be other devices that generate heat during operation. A heat spreading layer such as a layer of graphite may be attached to the backlight unit or other structures in the electronic device using adhesive. The adhesive may be patterned to form an unbonded area between at least some of the backlight unit or other structures to which the heat spreading layer is being attached and the heat spreading layer. The heat spreading layer may be mounted adjacent to a housing structure such as a metal midplate member that is attached to housing walls in the housing.