A new methodology is presented that allows the rapid determination of elastic constants of cubic fibre-textured thin films by X-ray diffraction. The theoretical concept is developed and tested on calculated examples of Cu and CrN films. The mechanical elastic constants are extrapolated from X-ray elastic constants by taking into consideration crystal and macroscopic elastic anisotropy. The derived algorithm enables the determination of a reflection and the corresponding value of the X-ray anisotropic factor Γ for which the X-ray elastic constants are equal to their mechanical counterparts in the case of fibre-textured cubic polycrystalline aggregates. The approach is independent of the crystal elastic anisotropy and depends on the fibre-texture type, the texture sharpness, the number of randomly oriented crystallites and the supposed grain-interaction model. In the experimental part, out-of-plane Young’s moduli of 111 and 311 fibre-textured Cu and CrN thin films deposited on monocrystalline Si(100) substrates are determined. The moduli are extrapolated from thin-film experimental X-ray elastic constants that are determined by a combination of X-ray diffraction substrate curvature and sin2ψ methods. For the calculation, the film macroscopic elastic anisotropy (texture) is considered. The advantage of the new technique lies in the fact that experimental moduli are determined nondestructively, using a static diffraction experiment, and represent volume-averaged quantities.