Background: The motivation for this study is the experimental evidence for rigid triaxial deformation at low energy in ^76Ge that was recently observed. Purpose: Quadrupole shapes and low-energy spectra of the isotopes ^72–82Ge are analyzed using a theoretical framework based on nuclear density functional theory. Method: The relativistic functional DD-PC1, supplemented by a finite-range pairing force, is used to perform constrained triaxial mean-field calculations of energy surfaces as functions of quadrupole deformation parameters. The corresponding collective Hamiltonian, based on DD-PC1, is employed in the calculation of excitation spectra and transition rates. Results: Model calculations reproduce the empirical trend of collective observables and predict the evolution of shapes from weakly triaxial in ^74Ge to γ soft in ^78,80Ge. For ^76Ge, in particular, the theoretical excitation spectrum is in good agreement with available data, the experimental ratio E(2^+_2)/E(2^+_1) is reproduced, as well as the pattern and amplitude of the staggering in energy between odd- and even-spin states in the γ band. Conclusions: The mean-field potential of ^76Ge appears to be γ soft. Collective correlations drive the nucleus toward triaxiality but do not stabilize a rigid triaxial shape. Both the experimental and theoretical staggering of levels in the γ band display a pattern consistent with triaxial shapes but the amplitudes are negligible and do not present evidence for rigid triaxiality.