We have studied the photoluminescence properties of GaN nanorods grown on Si(111) substrates by radio-frequency plasma-assisted molecular-beam epitaxy. The hexagonal shaped nanorods with lateral average diameters from 30 to 150 nm are obtained by controlling the Ga flux with a fixed amount of nitrogen. As the diameters decrease, the main emission lines assigned as donor bound excitons are blueshifted, causing a spectral overlap of this emission line with that of the free exciton at 10 K due to the quantum size effect in the GaN nanorods. The temperature-dependent photoluminescence spectra show an abnormal behaviour with an 'S-like' shape for higher diameter nanorods. The activation energy of the free exciton for GaN nanorods with different diameters was also evaluated.