Using Raman spectroscopy, we study the environmental sensitivity of mechanically exfoliated and electrically floating single-layer graphene transferred onto a hexagonal boron nitride (h-BN) substrate, in comparison with graphene deposited on a SiO(2) substrate. In order to understand and isolate the substrate effect on graphene electrical properties, we model and correct for Raman optical interference in the substrates. As-deposited and unannealed graphene shows a large I(2D)/I(G) ratio on both substrates, indicating extremely high quality, close to that of graphene suspended in vacuum. Thermal annealing strongly activates subsequent environmental sensitivity on the SiO(2) substrate; such activation is reduced but not eliminated on the h-BN substrate. In contrast, in a h-BN/graphene/h-BN sandwich structure, with graphene protected on both sides, graphene remains pristine despite thermal processing. Raman data provide a deeper understanding of the previously observed improved graphene electrical conductivity on h-BN substrates. In the sandwich structure, the graphene 2D Raman feature has a higher frequency and narrower line width than in pristine suspended graphene, implying that the local h-BN environment modestly yet measurably changes graphene electron and phonon dispersions.