When we listen to speech, our brain's neurophysiological responses "track" its acoustic features, but it is less well understood how these auditory responses are enhanced by linguistic content. Here, we recorded magnetoencephalography responses while subjects of both sexes listened to four types of continuous speechlike passages: speech envelope-modulated noise, English-like nonwords, scrambled words, and a narrative passage. Temporal response function (TRF) analysis provides strong neural evidence for the emergent features of speech processing in the cortex, from acoustics to higher-level linguistics, as incremental steps in neural speech processing. Critically, we show a stepwise hierarchical progression of progressively higher-order features over time, reflected in both bottom-up (early) and top-down (late) processing stages. Linguistically driven top-down mechanisms take the form of late N400-like responses, suggesting a central role of predictive coding mechanisms at multiple levels. As expected, the neural processing of lower-level acoustic feature responses is bilateral or right lateralized, with left lateralization emerging only for lexicosemantic features. Finally, our results identify potential neural markers, linguistic-level late responses, derived from TRF components modulated by linguistic content, suggesting that these markers are indicative of speech comprehension rather than mere speech perception.