Torques acting on galaxies lead to physical alignments, but the resulting ellipticity correlations are difficult to predict. As they constitute a major contaminant for cosmic shear studies, it is important to constrain the intrinsic alignment signal observationally. We measured the alignments of satellite galaxies within 90 massive galaxy clusters in the redshift range 0.05<z<0.55 and quantified their impact on the cosmic shear signal. We combined a sample of 38104 galaxies with spectroscopic redshifts with high-quality data from the Canada-France-Hawaii Telescope. We used phase-space information to select 14576 cluster members, 14250 of which have shape measurements and measured three different types of alignment: the radial alignment of satellite galaxies toward the brightest cluster galaxies (BCGs), the common orientations of satellite galaxies and BCGs, and the radial alignments of satellites with each other. Residual systematic effects are much smaller than the statistical uncertainties. We detect no galaxy alignment of any kind out to at least 3r_200_. The signal is consistent with zero for both blue and red galaxies, bright and faint ones, and also for subsamples of clusters based on redshift, dynamical mass, and dynamical state. These conclusions are unchanged if we expand the sample with bright cluster members from the red sequence. We augment our constraints with those from the literature to estimate the importance of the intrinsic alignments of satellites compared to those of central galaxies, for which the alignments are described by the linear alignment model. Comparison of the alignment signals to the expected uncertainties of current surveys such as the Kilo-Degree Survey suggests that the linear alignment model is an adequate treatment of intrinsic alignments, but it is not clear whether this will be the case for larger surveys.