Single-crystal diamond, with its unique optical, mechanical and thermal properties, has emerged as a promising material with applications in classical and quantum optics. However, the lack of heteroepitaxial growth and scalable fabrication techniques remains the major limiting factors preventing more wide-spread development and application of diamond photonics. In this work, we overcome this difficulty by adapting angled-etching techniques, previously developed for realization of diamond nanomechanical resonators, to fabricate racetrack resonators and photonic crystal cavities in bulk single-crystal diamond. Our devices feature large optical quality factors, in excess of 105, and operate over a wide wavelength range, spanning visible and telecom. These newly developed high-Q diamond optical nanocavities open the door for a wealth of applications, ranging from nonlinear optics and chemical sensing, to quantum information processing and cavity optomechanics.