The field of psycholinguistics is currently experiencing an explosion of interest in the analysis of neural oscillations - rhythmic brain activity synchronized at different temporal and spatial levels. Given that language comprehension relies on a myriad of processes, which are carried out in parallel in distributed brain networks, there is hope that this methodology might bring the field closer to understanding some of the more basic (spatially and temporally distributed, yet at the same time often overlapping) neural computations that support language function. In this review we discuss existing proposals linking oscillatory dynamics in different frequency bands to basic neural computations, and review relevant theories suggesting associations between band-specific oscillations and higher-level cognitive processes. More or less consistent patterns of oscillatory activity related to certain types of linguistic processing can already be derived from the evidence that has accumulated over the past few decades. The centerpiece of the current review is a synthesis of such patterns grouped by linguistic phenomenon. We restrict our review to evidence linking measures of oscillatory power to the comprehension of sentences, as well as linguistically (and/or pragmatically) more complex structures. For each grouping, we provide a brief summary and a table of associated oscillatory signatures that a psycholinguist might expect to find when employing a particular linguistic task. Summarizing across different paradigms, we conclude that a handful of basic neural oscillatory mechanisms are likely recruited in different ways and at different times for carrying out a variety of linguistic computations.