Pixels in an organic light-emitting diode (OLED) display may be microcavity OLED pixels having optical cavities. The optical cavities may be defined by a partially transparent cathode layer and a reflective anode structure. The anode of the pixels may include both the reflective anode structure and a supplemental anode that is transparent and that is used to tune the thickness of the optical cavity for each pixel. Organic light-emitting diode layers may be formed over the pixels and may have a uniform thickness in each pixel in the display. Pixels may have a conductive spacer between a transparent anode portion and a reflective anode portion, without an intervening dielectric layer. The conductive spacer may be formed from a material such as titanium nitride that is compatible with both anode portions. The transparent anode portions may have varying thicknesses to control the thickness of the optical cavities of the pixels.