We present phase-resolved spectroscopy of the He I 1.0830-mu m line in six Wolf-Rayet systems (WR 113 = CV Ser; WR 136 = HD 192163; WR 139 = V444 Cyg; WR 141 = HD 193928; WR 153 = GP Cep; and WR 155 = CQ Cep). Five of these systems are known WR+O binaries, with periods ranging from 1.6 d to nearly 30 d; WR 136 is only suspected of binarity. We find that the He I 1.0830-mu m line profile varies systematically with orbital phase, in a qualitatively similar manner for all systems (except WR 136). We interpret this variability as being a consequence of wind-wind interaction (i.e. colliding winds), and present results from simple models which include the effects of binary rotation and emission from the interaction region in a schematic manner. We find that the model qualitatively explains many (though nor all) characteristics of the observed variability, with the shock emission an important feature; we thereby demonstrate that variability in the He I 1.0830-mu m line is a sensitive indicator of wind dynamics in these colliding-wind systems.