We report strong exciton-photon coupling at room temperature in a hybrid high quality bulk 3 lambda/2 GaN cavity with a bottom lattice-matched AlInN/AlGaN distributed Bragg reflector through angle-resolved polarized photoluminescence (PL). Coupling of the optically active free excitons (X-A, X-B, and X-C) to the cavity mode is demonstrated, with their contribution to the PL spectra varying with polarization. Under TE polarization, exciton oscillator strengths for X-A and X-B are about one order of magnitude larger than in bulk GaAs. Photoluminescence exhibits a strong bottleneck effect despite its thermal lineshape.