We investigate the interaction of a graphene monolayer with the C(111) diamond surface using ab initio density functional theory. To accommodate the lattice mismatch between graphene and diamond, the overlayer deforms into a wavy structure that binds strongly to the diamond substrate. The detached ridges of the wavy graphene overlayer behave electronically as free-standing polyacetylene chains with delocalized π electrons, separated by regions containing only sp(3) carbon atoms covalently bonded to the (111) diamond surface. We performed quantum transport calculations for different geometries of the system to study how the buckling of the graphene layer and the associated bonding to the diamond substrate affect the transport properties. The system displays high carrier mobility along the ridges and a wide transport gap in the direction normal to the ridges. These intriguing, strongly anisotropic transport properties qualify the hybrid graphene-diamond system as a viable candidate for electronic nanodevices.