An electronic device such as a cover for a portable device may be provided with a body having hinge portions. The hinge portions may allow the body to bend about one or more bend axes. The cover may have electrical components such as a keyboard. A keyboard may be mounted at one end of the cover and a connector may be mounted at an opposing end of the cover. A flexible fabric signal path structure may be formed from metal traces on a flexible fabric substrate. At one end of the cover, the flexible fabric signal path structure may be coupled to a printed circuit in the keyboard using conductive adhesive. At the opposing end of the cover, the metal traces on the flexible fabric substrate may be coupled to the connector.