Propagation characteristics (resonances, propagation regions, and cutoffs) of ordinary waves (perpendicularly propagating electromagnetic waves) are studied in a relativistic electron plasma by using the kinetic model. The dispersion relation for the ordinary mode (O-mode) in a relativistic electron plasma is investigated by employing the Maxwell-Boltzmann-Jüttner distribution function. As the integration in the relativistic dispersion relation cannot be done analytically so an approximated value is obtained using the trapezoidal rule. Various modes of propagation for the ordinary waves are observed for each harmonic number n due to the relativistic effects defined by the value . We also observe that the high-temperature relativistic plasma environment is more transparent for the O-mode as compared to weakly relativistic and non-relativistic plasma environments. Moreover, the cutoff points are shifted to lower values in the relativistic limit.