Auditory rhythms are ubiquitous in music, speech, and other everyday sounds. Yet, it is unclear how perceived rhythms arise from the repeating structure of sounds. For speech, it is unclear whether rhythm is solely derived from acoustic properties (e.g., rapid amplitude changes), or if it is also influenced by the linguistic units (syllables, words, etc.) that listeners extract from intelligible speech. Here, we present three experiments in which participants were asked to detect an irregularity in rhythmically spoken speech sequences. In each experiment, we reduce the number of possible stimulus properties that differ between intelligible and unintelligible speech sounds and show that these acoustically-matched intelligibility conditions nonetheless lead to differences in rhythm perception. In Experiment 1, we replicate a previous study showing that rhythm perception is improved for intelligible (16-channel vocoded) as compared to unintelligible (1-channel vocoded) speech-despite near-identical broadband amplitude modulations. In Experiment 2, we use spectrally-rotated 16-channel speech to show the effect of intelligibility cannot be explained by differences in spectral complexity. In Experiment 3, we compare rhythm perception for sine-wave speech signals when they are heard as non-speech (for naïve listeners), and subsequent to training, when identical sounds are perceived as speech. In all cases, detection of rhythmic regularity is enhanced when participants perceive the stimulus as speech compared to when they do not. Together, these findings demonstrate that intelligibility enhances the perception of timing changes in speech, which is hence linked to processes that extract abstract linguistic units from sound.