Background: Neutron-rich nuclei with mass number between 100 and 110 attract much attention, since several kinds of shapes, such as spherical, prolate, oblate, and triaxial shapes, are predicted. In particular, for neutron-rich Mo isotopes, different models predict different magnitudes and rigidity of triaxial deformation. Previous interpretations of experimental results based solely on low-lying 22+ states are insufficient to distinguish between the rigid triaxial shape, γ vibration, or γ-soft rotor. Purpose: The shape evolution of Mo106, Mo108, and Mo110 is investigated through their 21+-state lifetimes, decay-branching ratios of the 22+ state, and energies of the low-lying collective excited states with Kπ=0+,2+, and 4+. Method: β-delayed γ-ray spectroscopy was employed for neutron-rich Nb and Zr isotopes produced at the RIKEN RI Beam Factory to populate excited states in Mo106, Mo108, and Mo110. The EUroball-RIKEN Cluster Array was used for high-resolution γ-ray detection and lifetimes of the 21+ states were determined using the UK fast-timing array of LaBr3(Ce) detectors. Results: New γ-ray transitions and levels are reported, including newly assigned 02+ states in Mo108,110. Quadrupole deformations were obtained for Mo106,108,110 from their 21+ energies and lifetimes. The β-delayed neutron-emission probabilities of Nb108 and Nb110 were determined by examining the γ rays of their respective daughter decays. Conclusions: The even-odd energy staggering in the 22+ band was compared with typical patterns of the γ-vibrational band, rigid triaxial rotor, and γ-soft rotor. The very small even-odd staggering of Mo106, Mo108, and Mo110 favors a γ-vibrational band assignment. The kinematic moment of inertia for the 22+ band showed a trend similar to the ground-state band, which is as expected for the γ-vibrational band. Beyond-mean-field calculations employing the constrained Hartree-Fock-Bogoliubov and local quasiparticle-random-phase approximation method using the SLy5+T interaction reproduced the ground and 22+ bands in Mo106 and Mo108. The collective wave functions are consistent with the interpretation of the 22+ band as the γ-vibrational band of the prolate shape. However, the staggering pattern observed in Mo110 differs from the one suggested in the calculations which predict a γ-soft rotor. There was no experimental indication of the oblate shape or the γ-soft rotor predicted in these Mo isotopes.