Photonic quantum technologies hold promise to repeat the success of integrated nanophotonic circuits in non-classical applications. Using linear optical elements, quantum optical computations can be performed with integrated optical circuits and can therefore overcome the existing limitations in terms of scalability. In addition to passive optical devices for realizing photonic quantum gates, active elements, such as single-photon sources and single-photon detectors, are essential ingredients for future optical quantum circuits. Material systems that allow for the monolithic integration of all components are particularly attractive, including III-V semiconductors, silicon and diamond. Here, we demonstrate nanophotonic integrated circuits made from high-quality polycrystalline diamond thin films in combination with on-chip single-photon detectors. By using superconducting nanowires that are coupled evanescently to traveling waves, we achieve high detection efficiencies of up to 66% as well as low dark count rates and a timing resolution of 190 ps. Our devices are fully scalable and hold promise for functional diamond photonic quantum devices.