The degradation of intrinsic properties of graphene during the transfer process constitutes a major challenge in graphene device fabrication, stimulating the need for direct growth of graphene on dielectric substrates. Previous attempts of metal-induced transformation of diamond and silicon carbide into graphene suffers from metal contamination and inability to scale graphene growth over large area. Here, we introduce a direct approach to transform polycrystalline diamond into high-quality graphene layers on wafer scale (4 inch in diameter) using a rapid thermal annealing process facilitated by a nickel, Ni thin film catalyst on top. We show that the process can be tuned to grow single or multilayer graphene with good electronic properties. Molecular dynamics simulations elucidate the mechanism of graphene growth on polycrystalline diamond. In addition, we demonstrate the lateral growth of free-standing graphene over micron-sized pre-fabricated holes, opening exciting opportunities for future graphene/diamond-based electronics.