The energy-momentum spectrum of the massive Nelson model, the Fröhlich polaron model and a large class of related models are known to satisfy an HVZ theorem. However, not much is rigorously known about the essential spectrum of these models. In this talk, some recent results including a Mourre estimate and absolute continuity of an interval just above the bottom of the essential spectrum will be presented. The results are proved in a common framework for a large class of massive translation invariant models and are valid for all values of the coupling constant.