An electronic device such as a wearable electronic device may have a band. The band may form a stand-alone device or a strap for a wristwatch unit or other device. Electrical components may be mounted on flexible printed circuit substrates. A substrate may be encapsulated by elastomeric polymer material or other material forming the band. The elastomeric polymer material may form cavities that receive the electrical components. Components such as light-emitting diodes may be mounted to the flexible printed circuit substrates so that the light-emitting diodes are located in the cavities. Reflective sidewalls in the cavities may reflect light from the light-emitting diodes outwardly through a thinned portion of the band. Light-diffusing material in the cavities may be formed from clear polymer with light-scattering particles.