The development of practical light sources based on group-IV semiconductors is a major outstanding goal of optoelectronics research, as a way to enable the continued integration of electronic and photonic functionalities on a CMOS compatible platform. However, this goal is severely complicated by the indirect energy bandgap of silicon, germanium, and related alloys. A possible solution is provided by the ability of biaxial tensile strain in Ge to lower the conduction-band edge at the direct (G) point relative to the L-valley minima, until at a strain of about 1.9% the fundamental bandgap becomes direct. Here we show that, by virtue of their ultrasmall thicknesses, Ge nanomembranes under externally applied mechanical stress are capable of accommodating such relatively high strain levels (up to over 2%). With this approach, we have demonstrated strong strain-enhanced Ge photoluminescence accompanied by a large strain-induced red shift in emission wavelength. A theoretical model of the emission properties of tensilely strained Ge has also been developed and applied to the measured high-strain luminescence spectra, providing evidence of population inversion. Finally, mechanically flexible photonic-crystal cavities have been developed on these nanomembranes, and used to demonstrate particularly large (20x) strain-induced enhancements in radiative efficiency, together with the observation of luminescence signatures associated with band-edge cavity modes. These results are promising for the development of group-IV semiconductor lasers for the technologically important short-wave infrared spectral region.