For users of cochlear implants (CIs), it can be exhaustingly difficult to focus on a speaker in an acoustically crowded environment because of suboptimal encoding of the audio signal captured by the CI microphones. Particularly, CI users are deprived of detailed monaural (rate pitch) and binaural (interaural time differences, ITDs) timing information usually supporting the cognitive task of voluntary stream segregation of different sound sources. We employed the Rhythmic Masking Release paradigm with bilaterally implanted listeners to study the effects of rate pitch, ITDs, and their combination on voluntary stream segregation with direct electric stimulation of listeners via a CI interface. The cues were applied to the target and distractor streams, potentially enabling the segregation of rhythmic patterns formed by the interleaved target and distractor signals. Our results indicate a general ability of CI listeners to segregate streams with a potential synergy effect of combining pitch and ITD cues. However, the rhythm segregation based on ITDs alone seems to be challenging, probably because CI users are not used to access salient ITDs in everyday life. This suggests that training may be important for exploiting sound segregation cues provided by future bilateral CI systems, particularly regarding ITDs.