Pyramidal GaAs structures on top of GaAs/AlAs distributed Bragg reflectors are investigated as candidates for true three-dimensional cavities with potentially low mode volume and high quality-factor. Different types of single and coupled resonators with base lengths of a few microns are realized using a combination of molecular-beam epitaxy, electron-beam lithography, and wet chemical etching. Embedded InGaAs quantum dots are utilized as light sources to verify the resonator modes. Furthermore, a spatially localized emission through the pyramid facets indicates the future possibility of coupling cavity modes to optical fibers. This could be interesting within the context of single photon emitters.