A fully three-dimensional (3D) one-step late (OSL), maximum a posteriori (MAP) reconstruction algorithm based on the median root prior (MRP) was implemented and evaluated for the reconstruction of 3D positron emission tomography (PET) studies. The algorithm uses the ordered subsets (OS) scheme for convergence acceleration and data update during iterations. The algorithm was implemented using the software package developed within the EU project PARAPET (www.brunel.ac.uk/~masrppet). The MRP algorithm was evaluated using experimental phantom and real 3D PET brain studies. Various experimental set-ups in terms of activity distribution and counting statistics were considered. The performance of the algorithm was assessed by calculating figures of merit such as: contrast, coefficient of variation, activity ratio between two regions and full width at half of maximum for resolution measurements. The performance of MRP was compared with that of 3D ordered subsets-expectation maximisation (OSEM) and 3D re-projection (3DRP) algorithms. In all the experimental situations considered, MRP showed: (1) convergence to a stable solution, (2) effectiveness in noise reduction, particularly for low statistics data, (3) good preservation of spatial details. Compared with the OSEM and 3DRP algorithms, MRP provides comparable or better results depending on the parameters used for the reconstruction of the images.