An electronic device may have a layer of material with transparent portions. The layer of material may be fabric layer, a layer of leather, or other layer of material. The transparent portions may be formed from openings in the layer of material. The transparent portions may overlap electrical components. The electrical components may include a sensor, light-emitting components, and haptic output components. The light-emitting components may include light-emitting diodes, edge-lit light guides, displays having arrays of pixels, or other light-emitting devices. The openings in the layer of material may be patterned to form icon shapes and other shapes. The layer of material may be used in forming a strap for a wristwatch, a case for a cellular telephone or other portable device, or other electronic device.