We report micro-photoluminescence studies of single GaN/AlN quantum dots grown along the (0001) crystal axis by molecular beam epitaxy on Si(111) substrates. The emission lines exhibit a linear polarization along the growth plane, but with varying magnitudes of the polarization degree and with principal polarization axes that do not necessarily correspond to crystallographic directions. Moreover, we could not observe any splitting of polarized emission lines, at least within the spectral resolution of our setup (1 meV). We propose a model based on the joint effects of electron-hole exchange interaction and in-plane anisotropy of strain and/or quantum dot shape, in order to explain the quantitative differences between our observations and those previously reported on, e.g. CdTe- or InAs-based quantum dots.