We study the effects of polarizable substrates such as SiO2 and SiC on the carrier dynamics in graphene. We find that the quasiparticle spectrum acquires a finite broadening due to the long-range interaction with the polar modes at the interface between graphene and the substrate. This mechanism results in a density dependent electrical resistivity, that exhibits a sharp increase around room temperature, where it can become the dominant limiting factor of electron transport. The effects are weaker in doped bilayer graphene, due to the more conventional parabolic band dispersion.