The γ-ray strength functions and level densities of Ge73,74 have been extracted up to the neutron-separation energy Sn from particle-γ coincidence data using the Oslo method. Moreover, the γ-ray strength function of Ge74 above Sn has been determined from photoneutron measurements; hence these two experiments cover the range of Eγ≈1-13 MeV for Ge74. The obtained data show that both Ge73,74 display an increase in strength at low γ energies. The experimental γ-ray strength functions are compared with M1 strength functions deduced from average B(M1) values calculated within the shell model for a large number of transitions. The observed low-energy enhancements in Ge73,74 are adopted in the calculations of the Ge72,73(n,γ) cross sections, where there are no direct experimental data. Calculated reaction rates for more neutron-rich germanium isotopes are shown to be strongly dependent on the presence of the low-energy enhancement.