Organic light-emitting diodes (OLEDs) suffer from poor light outcoupling efficiency (ηout < 20%) due to large internal waveguiding in the high-index layers/substrate, and plasmonic losses at the metal cathode interface. A promising approach to enhance light outcoupling is to utilize internal periodic corrugations that can diffract waveguided and plasmonic modes back to the air cone. Although corrugations can strongly diffract trapped modes, the optimal geometry of corrugations and limits to ηout are not well-established. We develop a general rigorous scattering matrix theory for light emission from corrugated OLEDs, by solving Maxwell’s equations in Fourier space, incorporating the environment-induced modification of the optical emission rate (Purcell effect). We computationally obtain the spectrally emissive power inside and outside the OLED. We find conformally corrugated OLEDs, where all OLED interfaces are conformal with a photonic crystal substrate, having triangular lattice symmetry, exhibit high light outcoupling ηout ∼60–65%, and an enhancement factor exceeding 3 for optimal pitch values between 1 and 2.5 μm. Waveguided and surface plasmon modes are strongly diffracted to the air cone through first-order diffraction. ηout is insensitive to corrugation heights larger than 100 nm. There is a gradual roll-off in ηout for a larger pitch and sharper decreases for small pitch values. Plasmonic losses remain below 10% for all corrugation pitch values. Our predicted OLED designs provide a pathway for achieving very high light outcoupling over the full optical spectrum that can advance organic optoelectronic science and solid-state lighting.