Hybrid circuits combining traditional nanophotonic components with carbon-based materials are emerging as a promising platform for optoelectronic devices. We demonstrate such circuits by integrating single-layer graphene films with silicon nitride waveguides as a new architecture for broadband optical operation. Using high-quality microring resonators and Mach-Zehnder interferometers with extinction ratios beyond 40 dB we realize flexible circuits for phase-sensitive detection on chip. Hybrid graphene-photonic devices are fabricated via mechanical transfer and lithographic structuring, allowing for prolonged light-matter interactions. Our approach holds promise for studying optical processes in low-dimensional physical systems and for realizing electrically tunable photonic circuits.