Recent evidence suggests that visuospatial attention is not sustained but fluctuates in a rhythmic fashion, sampling location after location in each cycle of this rhythm. However, studies differ in which location rhythmic attention becomes evident: the cued, non-cued location or both. It is also unclear how the behavioral relevance of each sampled location determines the rhythmic sampling patterns. Here, we aim to elucidate these two issues. First, we aim to find evidence of rhythmic attention at the predicted (i.e. cued) location under moderately informative predictor value, replicating earlier studies. Second, we hypothesize that rhythmic attentional sampling behavior will be affected by the behavioral relevance of the sampled location, ranging from non-informative to fully informative. To these aims, we used a modified Egly-Driver task with three conditions: a fully informative cue, a moderately informative cue (replication condition), and a non-informative cue. We did not find evidence of rhythmic sampling at cued locations, failing to replicate earlier studies. However, we did find attentional rhythmicity at non-cued locations, specifically on different objects. This effect was only present when the cue was informative, not when the cue was non-informative. These results suggest that the strongest evidence of rhythmic sampling can be found at spatial locations that bear relatively less behavioral relevance. This suggests an occasional dis-engagement mechanism, whereby attention regularly switches from behaviorally relevant locations to less behaviorally relevant locations.