In the present study, we investigated if individuals with neurogenic speech sound impairments of three types, Parkinson’s dysarthria, apraxia of speech, and aphasic phonological impairment, accommodate their speech to the natural speech rhythm of an auditory model, and if so, whether the effect is more significant after hearing metrically regular sentences as compared to those with an irregular pattern. This question builds on theories of rhythmic entrainment, assuming that sensorimotor predictions of upcoming events allow humans to synchronize their actions with an external rhythm. To investigate entrainment effects, we conducted a sentence completion task relating participants’ response latencies to the spoken rhythm of the prime heard immediately before. A further research question was if the perceived rhythm interacts with the rhythm of the participants’ own productions, i.e., the trochaic or iambic stress pattern of disyllabic target words. For a control group of healthy speakers, our study revealed evidence for entrainment when trochaic target words were preceded by regularly stressed prime sentences. Persons with Parkinson’s dysarthria showed a pattern similar to that of the healthy individuals. For the patient groups with apraxia of speech and with phonological impairment, considerably longer response latencies with differing patterns were observed. Trochaic target words were initiated with significantly shorter latencies, whereas the metrical regularity of prime sentences had no consistent impact on response latencies and did not interact with the stress pattern of the target words to be produced. The absence of an entrainment in these patients may be explained by the more severe difficulties in initiating speech at all. We discuss the results in terms of clinical implications for diagnostics and therapy in neurogenic speech disorders.