While stickiness and power-law behavior of Poincare recurrence statistics for two-degree-of-freedom systems is well-understood, this remains to be an open problem for higher-dimensional systems. We study such intermittent behavior of chaotic trajectories in three-dimensional volume-preserving systems using the example of the Arnold-Beltrami-Childress map. If two action-like variables of the map are nearly conserved, the configuration space displays tubular regular structures surrounded by a chaotic sea. Stickiness occurs around these tubes and manifestations of partial barriers to transport are observed. Our investigation of the dynamics in frequency space indicates that coupled resonances play a crucial role.