One important attribute of graphene that makes it attractive for high-performance electronics is its inherently large thermal conductivity (κ) for the purposes of thermal management. Using a combined density-functional theory and classical molecular-dynamics approach, we predict that the κ of graphene supported on hexagonal boron nitride (h-BN) can be as large as 90% of the κ of suspended graphene, in contrast to the significant suppression of κ (more than 70% reduction) on amorphous silica. Interestingly, we find that this enhanced thermal transport is largely attributed to increased lifetimes of the in-plane acoustic phonon modes, which is a notable contrast from the dominant contribution of out-of-plane acoustic modes in suspended graphene. This behavior is possible due to the charge polarization throughout graphene that induces strong interlayer adhesion between graphene and h-BN. These findings highlight the potential benefit of layered dielectric substrates such as h-BN for graphene-based thermal management, in addition to their electronic advantages. Furthermore, our study brings attention to the importance of understanding the interlayer interactions of graphene with layered dielectric materials which may offer an alternative technological platform for substrates in electronics.