The propagation and absorption of ordinary waves in fully relativistic plasma are investigated. Several interesting effects introduced by the relativistic mass increase associated with an increase in the electron temperature are examined. In particular, it is found that the cutoff density of ordinary waves increases significantly as the electron temperature increases. The effects of anomalous wave dispersion, which appear near the fundamental electron cyclotron frequency and its second and third harmonic, are greatest for low temperatures and subsist up to intermediate temperatures ( keV). An efficient damping of ordinary waves is obtained in a wide temperature range. The harmonic structure of the absorption profiles and the effects of anomalous dispersion disappear almost simultaneously at high electron temperatures. The results obtained may have important implications for the wave absorption and in particular, for emission in relativistic plasmas.