DNS have been performed of turbulent channel flow with spanwise rotation and bulk Reynolds number Re up to 30000. At moderate rotation rates Ro the flow on one side of the channel is approximately laminar, or has turbulent patches or oblique turbulent-laminar patterns. Intense cyclic bursts of turbulence with long time intervals are in some cases observed at sufficiently high Re and Ro. A linear analysis indicates that the turbulence bursts are initiated by a linear instability of plane waves aligned with the rotation axis. This linear instability can develop even if parts of the flow are strongly turbulent.