The dipole strength of the nuclide $^{76}$Ge was studied in photon-scattering experiments using bremsstrahlung produced with electron beams of energies of 7.8 and 12.3 MeV at the $\gamma$ELBE facility. We identified 210 levels up to an excitation energy of 9.4 MeV and assigned spin $J$ = 1 to most of them. The quasicontinuum of unresolved transitions was included in the analysis of the spectra and the intensities of branching transitions were estimated on the basis of simulations of statistical $\gamma$-ray cascades. The photoabsorption cross section up to the neutron-separation energy was determined and is compared with predictions of the statistical reaction model. The derived photon strength function is compared with results of experiments using other reactions.