Nanomechanical resonators realized from tensile-strained materials reach ultralow mechanical dissipation in the kHz to MHz frequency range. Tensile-strained crystalline materials that are compatible with epitaxial growth of heterostructures would thereby at the same time allow realizing monolithic free-space optomechanical devices, which benefit from stability, ultrasmall mode volumes, and scalability. In our work, we demonstrate nanomechanical string and trampoline resonators made from tensile-strained InGaP, which is a crystalline material that is epitaxially grown on an AlGaAs heterostructure. We characterize the mechanical properties of suspended InGaP nanostrings, such as anisotropic stress, yield strength, and intrinsic quality factor. We find that the latter degrades over time. We reach mechanical quality factors surpassing 107 at room temperature with a Q·f product as high as 7 × 1011Hz with trampoline-shaped resonators. The trampoline is patterned with a photonic crystal to engineer its out-of-plane reflectivity, desired for efficient signal transduction of mechanical motion to light.