We report on the design, fabrication and measurement of travelling-wave superconducting nanowire single-photon detectors (SNSPDs) integrated with polycrystalline diamond photonic circuits. We analyze their performance both in the near-infrared wavelength regime around 1600 nm and at 765 nm. Near-IR detection is important for compatibility with the telecommunication infrastructure, while operation in the visible wavelength range is relevant for compatibility with the emission line of silicon vacancy centers in diamond which can be used as efficient single-photon sources. Our detectors feature high critical currents (up to 31 {mu}A) and high performance in terms of efficiency (up to 74% at 765 nm), noise-equivalent power (down to 4.4*10^-19 W/(Hz^1/2) at 765 nm) and timing jitter (down to 23 ps).