A display may have thin-film transistor circuitry on a substrate. An array of organic light-emitting diodes may be formed on the thin-film transistor circuitry. The organic light-emitting diodes may have anodes, cathodes, and emissive material located between the anodes and cathodes. A circular polarizer may be formed over the array of organic light-emitting diodes. The circular polarizer may include a linear polarizer and a quarter wave plate. The linear polarizer may be formed from one or more film layers having narrowband dichroic dyes so that the polarizer exhibits transmission peaks aligned with a selected subset of wavelengths and absorbance notches corresponding to the selected subset of wavelengths. The selected subset of wavelengths may cover the ranges where the light-emitting diodes are outputting light. Configured in this way, the polarizer will exhibit enhanced luminance at the desired wavelengths while suppressing ambient light reflections at other wavelengths in the visible spectrum.