A display may have an array of organic light-emitting diode (OLED) pixels that each have OLED layers interposed between a cathode and an anode. Voltage may be applied to the anode of each pixel to control the magnitude of emitted light. The conductivity of the OLED layers may allow leakage current to pass between neighboring anodes in the display. To reduce leakage current and cross-talk, the thickness of at least one of the OLED layers may be reduced. To maintain the optical cavity of the pixels, transparent optical spacer structures may be inserted. Alternatively, the thickness of the anodes can be increased. To accommodate a common prime layer within the OLED layers, the optical spacers or anodes may be separately patterned to have different thicknesses. Grating structures and photonic crystal structures may be embedded as part of the optical spacers to help control emission at selected viewing angles.