Background: Predictions of spectroscopic properties of low-lying states are critical for nuclear structure studies. Theoretical methods can be particularly involved for odd-mass nuclei because of the interplay between the unpaired nucleon and collective degrees of freedom. Only a few models have been developed for systems in which octupole collective degrees of freedom play a role. Purpose: We aim to predict spectroscopic properties of odd- mass nuclei characterized by octupole shape deformation, employing a model that describes single- particle and collective degrees of freedom within the same microscopic framework. Method: A microscopic core-quasiparticle coupling (CQC) model based on the covariant density functional theory is developed, which includes collective excitations of even-mass core nuclei and single-particle states of the odd nucleon, calculated using a quadrupole-octupole collective Hamiltonian combined with a constrained reflection-asymmetric relativistic Hartree-Bogoliubov model. Results: Model predictions for low-energy excitation spectra and transition rates of odd-mass radium isotopes 223, 225, 227Ra are shown to be in good agreement with available data. Conclusions: A microscopic CQC model based on covariant density functional theory has been developed for odd-mass nuclei characterized by both quadrupole and octupole shape deformations. Theoretical results reproduce data in odd-mass Ra isotopes and provide useful predictions for future studies of octupole correlations in nuclei and related phenomena.