III-V semiconductor quantum dots (QDs) are near-ideal and versatile single-photon sources. Because of the capacity for monolithic integration with photonic structures as well as optoelectronic and optomechanical systems, they are proving useful in an increasingly broad application space. Here, we develop monolithic circular dielectric gratings on bulk substrates - as opposed to suspended or wafer-bonded substrates - for greatly improved photon collection from InAs quantum dots. The structures utilize a unique two-tiered distributed Bragg reflector (DBR) structure for vertical electric field confinement over a broad angular range. Opposing "openings" in the cavities induce strongly polarized QD luminescence without harming collection efficiencies. We describe how measured enhancements depend on the choice of collection optics. This is important to consider when evaluating the performance of any photonic structure that concentrates farfield emission intensity. Our cavity designs are useful for integrating QDs with other quantum systems that require bulk substrates, such as surface acoustic wave phonons.