Motion in depth can be perceived from binocular cues alone, yet it is unclear whether these cues support speed sensitivity in the absence of the monocular cues that normally co-occur in natural viewing. We measure threshold contours in space-time for the discrimination of three-dimensional (3D) motion to determine whether observers use speed to discriminate a test 3D motion from two identical standards. We compare thresholds for random-dot stereograms (RDS) containing both binocular cues to 3D motion-interocular velocity difference and changing disparity over time-with performance for dynamic random-dot stereograms (DRDS), which contain only the second cue. Threshold contours are tilted along the axis of constant velocity in space-time for RDS stimuli at slow speeds (0.5 m/s), evidence for speed sensitivity. However, for higher speeds (1.5 m/s) and DRDS stimuli, observers rely on the component cues of duration and disparity. In a second experiment, noise of constant velocity is added to the standards to degrade the reliability of these separate components. Again there is evidence for speed tuning for RDS, but not for DRDS. Considerable variation is observed in the ability of individual observers to use the different cues in both experiments, however, in general the results emphasize the importance of interocular velocity difference as a critical cue for speed sensitivity to motion in depth, and suggest that speed sensitivity to stereomotion from binocular cues is restricted to relatively slow speeds.