Highly ordered two-dimensional self-organized nanochannel networks as well as free-standing nanomembranes are produced on rigid substrates by means of III-V semiconductor compressively strained layers grown on top of an etchant-sensitive material. The releasing process is controlled by regularly spaced pits obtained from photolithography and a subsequent wet chemical etching. By tuning basic film parameters such as strain and thickness, one obtains periodic arrays of two-dimensional nanochannel networks with symmetries defined by the shape and periodicity of the photolithographic starting pits. Such nanochannel networks with a submicroscale lateral feature size exhibit a surprising flexibility with respect to the crystal lattice symmetry, retaining the original film crystalline quality as confirmed by X-ray grazing-incidence diffraction (GID) measurements. Finite element modeling helps in understanding the particular process of the cross-nanochannel formation.