According to the popular Grounded Cognition Model (GCM), the sensory and motor features of concepts, including word meanings, are stored directly within neural systems for perception and action. More precisely, the core claim is that these concrete conceptual features reuse some of the same modality‐specific representations that serve to categorize experiences involving the relevant kinds of objects and events. Research in semantic typology, however, has shown that word meanings vary significantly across the roughly 6500 languages in the world. I argue that this crosslinguistic semantic diversity has significant yet previously unrecognized theoretical consequences for the GCM. In particular, to accommodate the typological data, the GCM must assume that the concrete features of word meanings are not merely stored within sensory/motor brain systems, but are represented there in ways that are, to a nontrivial degree, language‐specific. Moreover, it must assume that these conceptual representations are also activated during the nonlinguistic processing of the relevant kinds of objects and events (e.g., during visual perception and action planning); otherwise, they would not really be grounded, which is to say, embedded inside sensory/motor systems. Crucially, however, such activations would constitute what is traditionally called linguistic relativity—that is, the influence of language‐specific semantic structures on other forms of cognition. The overarching aim of this paper is to elaborate this argument more fully and explore its repercussions. To that end, I discuss in greater detail the key aspects of the GCM, the evidence for crosslinguistic semantic diversity, pertinent work on linguistic relativity, the central claim that the GCM entails linguistic relativity, some initial supporting results, and some important limitations and future directions.