Auditory selective attention makes it possible to pick out one speech stream that is embedded in a multispeaker environment. We adapted a cued dichotic listening task to examine suppression of a speech stream lateralized to the nonattended ear, and to evaluate the effects of attention on the right ear's well‐known advantage in the perception of linguistic stimuli. After being cued to attend to input from either their left or right ear, participants heard two different four‐word streams presented simultaneously to the separate ears. Following each dichotic presentation, participants judged whether a spoken probe word had been in the attended ear's stream. We used EEG signals to track participants' spatial lateralization of auditory attention, which is marked by interhemispheric differences in EEG alpha (8–14 Hz) power. A right‐ear advantage (REA) was evident in faster response times and greater sensitivity in distinguishing attended from unattended words. Consistent with the REA, we found strongest parietal and right frontotemporal alpha modulation during the attend‐right condition. These findings provide evidence for a link between selective attention and the REA during directed dichotic listening.