Bright single-photon emission from single quantum dots (QDs) in suspended circular Bragg grating microcavities is demonstrated. This geometry has been designed to achieve efficient (> 50%) single-photon extraction into a near-Gaussian-shaped far-field pattern, modest (≈10x) Purcell enhancement of the radiative rate, and a spectral bandwidth of a few nanometers. Measurements of fabricated devices show progress toward these goals, with collection efficiencies as high as ≈ 10% demonstrated with moderate spectral bandwidth and rate enhancement. Photon correlation measurements are performed under above-bandgap excitation (pump wavelength = 780 to 820 nm) and confirm the single-photon character of the collected emission. While the measured sources are all antibunched and dominantly composed of single photons, the multiphoton probability varies significantly. Devices exhibiting tradeoffs among collection efficiency, Purcell enhancement, and multiphoton probability are explored and the results are interpreted with the help of finite-difference time-domain simulations. Below-bandgap excitation resonant with higher states of the QD and/or cavity (pump wavelength = 860 to 900 nm) shows a near-complete suppression of multiphoton events and may circumvent some of the aforementioned tradeoffs.