GaN-on-diamond is a promising route towards reliable high-power transistor devices with outstanding performances due to better heat management, replacing common GaN-on-SiC technologies. Nevertheless, the implementation of GaN-on-diamond remains challenging. In this work, the selective area growth of GaN nanostructures on cost-efficient, large-scale available heteroepitaxial diamond (001) substrates by means of plasma-assisted molecular beam epitaxy is investigated. Additionally, we discuss the influence of an AlN buffer on the morphology of the GaN nanostructures. The nanowires and nanofins are characterized by a very high selectivity and controllable dimensions. Low temperature photoluminescence measurements are used to evaluate their structural quality. The growth of two GaN crystal domains, which are in-plane rotated against each other by 30°, is observed. The favoring of a certain domain is determined by the off-cut direction of the diamond substrates. By X-ray diffraction we show that the GaN nanostructures grow perpendicular to the diamond surface on off-cut diamond (001) substrates, which is in contrast to the growth on diamond (111), where the nanostructures are aligned with the substrate lattice. Polarity-selective wet chemical etching and Kelvin probe force microscopy reveal that the GaN nanostructures grow solely in the Ga-polar direction. This is a major advantage compared to the growth on diamond (111) and enables the application of GaN nanostructures on cost-efficient diamond for high-power/high-frequency applications.