Fabrication of microcavities based on III-nitrides is challenging due to difficulties with the coherent growth of heterostructures having a large number of periods, at the same time keeping a good precision in terms of thickness and composition of the alloy. A planar design for GaN microresonators supporting whispering gallery modes is suggested. GaN hexagonal microstructures are fabricated by selective-area metalorganic vapor phase epitaxy using focused ion beam for mask patterning. Low-temperature cathodoluminescence spectra measured with a high spatial resolution demonstrate two dominant emission lines in the near bandgap region. These lines merge at room temperature into a broad emission band peaking at approximate to 3.3 eV, which is shifted toward lower energies compared with the reference excitonic spectrum measured for the GaN layer. A numerical analysis of exciton-polariton modes shows that some strongly localized cavity modes can have high Purcell coefficients and can strongly interact with the GaN exciton.