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