Embodied cognition--the idea that mental states and processes should be understood in relation to one's bodily constitution and interactions with the world--remains a controversial topic within cognitive science. Recently, however, increasing interest in predictive processing theories among proponents and critics of embodiment alike has raised hopes of a reconciliation. This article sets out to appraise the unificatory potential of predictive processing, focusing in particular on embodied formulations of "active inference." Our analysis suggests that most active-inference accounts invoke weak, potentially trivial conceptions of embodiment; those making stronger claims do so independently of the theoretical commitments of the active-inference framework. We argue that a more compelling version of embodied active inference can be motivated by adopting a diachronic perspective on the way rhythmic physiological activity shapes neural development in utero. According to this "visceral afferent training" hypothesis, early-emerging physiological processes are essential not only for supporting the biophysical development of neural structures but also for configuring the cognitive architecture those structures entail. Focusing in particular on the cardiovascular system, we propose three candidate mechanisms through which visceral afferent training might operate: (a) activity-dependent neuronal development, (b) periodic signal modeling, and (c) oscillatory network coordination.