We have investigated the characteristics of exciton-photon strong coupling in a CuBr bulk microcavity that consists of a CuBr active layer with an effective thickness of lambda/2 and HfO2/SiO2 distributed Bragg reflectors: lambda corresponds to an effective resonant wavelength of the lowest-lying exciton. The CuBr crystal has three excitons labeled Z(f), Z(1,2), and Z(3) at the Gamma point, where the Z(f) exciton originates from a triplet state, which is peculiar to CuBr. Angle-resolved reflectance spectra measured at 10 K demonstrate the strong coupling behavior of the Z(f), Z(1,2), and Z(3) excitons and cavity photon, resulting in the formation of four cavity-polariton branches. Analyzing the cavity-polariton dispersion relations based on a phenomenological Hamiltonian for the strong coupling, we evaluated the vacuum Rabi-splitting energies of the Z(f), Z(1,2), and Z(3) excitons to be 31, 108, and 84 meV, respectively. These Rabi-splitting energies reflect the magnitudes of the oscillator strengths of the relevant excitons. Furthermore, we precisely measured angle-resolved photoluminescence (PL) spectra of the lower polariton branch under a weak excitation condition. In the bottleneck region, the population of the cavity polaritons is negligible, and the PL intensity at k = 0 is the highest. These facts suggest that the relaxation process of the cavity polaritons is not affected by a bottleneck effect.