Some inaccuracies are possible during the near-surface residual stress measurement of textured material by X-ray diffraction using the standard multiexposure technique. Diffraction intensities depend on factors like inclination of the sample, the rotation of the beam, and the position of detectors. Low intensity due to texture can result in increased percentage error observed stress values. Anisotropy of elastic constants in textured materials can further increase the errors in measured stress. In the present study, the effect of texture on residual stress has been studied for seamless Ti-3wt%Al-2Vwt% tubes, in pilgered and annealed conditions, by superimposing the residual stress measurements on the pole figure obtained by texture measurement. A general approach has been developed for accurate determination of the stress tensor in textured material. This consists of the selection of inclinations giving rise to high intensity peak, the selection of rotation angles symmetric to the pole figure to get similar intensity from both the detectors (where this is not possible one detector with low intensity is shut off), and the calculation of the bulk X-ray elastic constants using the single crystal elastic data and texture. Cold worked Ti-3wt%Al-2Vwt% tubes had the highest stress along the rolling direction and that is compressive in nature. The principal residual stresses in the pilgered tubes were along the rolling and transverse directions, and annealing leads to reduction and reorientation of the stresses.