We demonstrate the fabrication of SiGe nanomembranes (NM): fully elastically relaxed, smooth, single-crystalline sheets of SiGe alloy. A thin SiGe layer is grown on a silicon-on-insulator (SOI) substrate with molecular beam epitaxy, followed by a Si capping layer with thickness similar to that of the Si template layer of the SOI. The SiO2 layer of the SOI is selectively etched away, leaving the Si/SiGe/Si trilayer heterostructure free to strain share. The Si layers of the trilayer are then selectively etched away, leaving a fully elastically relaxed SiGe NM. These SiGe NMs are then transferred to new handling substrates and bonded. The strain states of the SiGe NMs are measured throughout the fabrication process with Raman spectroscopy. Initially, the SiGe is fully strained to the Si lattice constant, and relaxes to the bulk SiGe lattice constant only after release from the original growth substrate and removal of the surrounding Si layers.