Inspired by the recent experimental data [J.-G. Wang, et al., Phys. Lett. B 675 (2009) 420], we extend the triaxial projected shell model approach to study the γ-band structure in odd-mass nuclei. As a first application of the new development, the γ-vibrational structure of 103Nb is investigated. It is demonstrated that the model describes the ground-state band and multi-phonon γ-vibrations quite satisfactorily, supporting the interpretation of the data as one of the few experimentally-known examples of simultaneous occurrence of one- and two-γ-phonon vibrational bands. This generalizes the well-known concept of the surface γ-oscillation in deformed nuclei built on the ground-state in even–even systems to γ-bands based on quasiparticle configurations in odd-mass systems.