In organic field-effect transistors (FETs), charges move near the surface of an organic semiconductor, at the interface with a dielectric. In the past, the nature of the microscopic motion of charge carriers--which determines the device performance--has been related to the quality of the organic semiconductor. Recently, it was discovered that the nearby dielectric also has an unexpectedly strong influence. The mechanisms responsible for this influence are not understood. To investigate these mechanisms, we have studied transport through organic single-crystal FETs with different gate insulators. We find that the temperature dependence of the mobility evolves from metallic-like to insulating-like with increasing dielectric constant of the insulator. The phenomenon is accounted for by a two-dimensional Fröhlich polaron model that quantitatively describes our observations and shows that increasing the dielectric polarizability results in a crossover from the weak to the strong polaronic coupling regime. This represents a considerable step forward in our understanding of transport through organic transistors, and identifies a microscopic physical process with a large influence on device performance.