We report on a theoretical study of free-standing phase transmission gratings for high-resolution extreme ultraviolet (EUV) and soft x-ray spectroscopy and investigate their properties. Designed for wavelengths between about 2 and 40 nm, the devices may provide a first order diffraction efficiency beyond 30%. We use rigorous coupled wave analysis methods in order to optimize the grating design parameters and discuss features of segmented grating arrays. Elemental, as well as compound, materials such as Be, Mo, LiF, and poly-(methylmethacrylate) are considered with respect to their potential and practical limitations in terms of feasibility and sensitivity to radiation damage. Simulations are performed for several samples on the radiation produced by a table-top EUV plasma source and applications to astrophysical problems are considered.