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The results of theoretical and experimental studies of X-ray emission from relativistic electrons crossing periodic nanostructures are presented in this work. Simple formulae describing the characteristics of the total emission from either thin-non-absorbing or thick-absorbing multilayers are derived, accounting for both diffracted-transition and parametric-radiation mechanisms. The measured vertical spatial distributions and orientation dependencies (rocking curves) of the spectra of this type of radiation are presented. Good agreement between theory and experimental results is shown. The results demonstrate the possibility of a tunable quasimonochromatic X-ray source whose efficiency can be larger than that of other novel X-ray sources