Asymmetry in auditory cortical oscillations could play a role in speech perception by fostering hemispheric triage of information across the two hemispheres. Due to this asymmetry, fast speech temporal modulations relevant for phonemic analysis could be best perceived by the left auditory cortex, while slower modulations conveying vocal and paralinguistic information would be better captured by the right one. It is unclear, however, whether and how early oscillation-based selection influences speech perception. Using a dichotic listening paradigm in human participants, where we provided different parts of the speech envelope to each ear, we show that word recognition is facilitated when the temporal properties of speech match the rhythmic properties of auditory cortices. We further show that the interaction between speech envelope and auditory cortices rhythms translates in their level of neural activity (as measured with fMRI). In the left auditory cortex, the neural activity level related to stimulus–brain rhythm interaction predicts speech perception facilitation. These data demonstrate that speech interacts with auditory cortical rhythms differently in right and left auditory cortex, and that in the latter, the interaction directly impacts speech perception performance.