We investigate the residual strains in a free-standing Cu/Nb multilayer of 30 nm nominal layer thickness with synchrotron x-rays. This material system is characterized by columnar grains of Cu and Nb with incoherent interfaces and a sharp physical-vapor-deposition texture. High energy x-rays were used with an area detector along with multiple sample rotations to yield diffraction strain components in a very large number of directions. Due to the texture and the elastic anisotropy of constituents, observed diffraction strains cannot be derived from a single strain tensor (also known as linear sin2 ψ). Orientation-dependent diffraction strain modeling is utilized with a Vook-Witt micromechanical model. Obtained phase-resolved in-plane stress magnitudes are -515 MPa in Nb and +513 MPa in Cu, satisfying force equilibrium within experimental errors. The stresses of this magnitude will certainly influence the mechanical behavior of the multilayer upon further loading. The Vook-Witt model describes the Nb diffraction strains very well, and thereby provides information on the stress distribution in crystallites as a function of their orientation. On the other hand, the same level of agreement with the Vook-Witt model has not been achieved for Cu diffraction strains.