A fast iterative method is described for processing clinical PET scans acquired in three dimensions, that is, with no inter-plane septa, using standard computers to replace dedicated processors used until the late 1990s. The method is based on sinogram resampling, Fourier rebinning, Monte Carlo scatter simulation and iterative reconstruction using the attenuation-weighted OSEM method and a projector based on a Gaussian pixel model. Resampling of measured sinogram values occurs before Fourier rebinning, to minimize parallax and geometric distortions due to the circular geometry, and also to reduce the size of the sinogram. We analyse the geometrical and statistical effects of resampling, showing that the lines of response are positioned correctly and that resampling is equivalent to about 4 mm of post-reconstruction filtering. We also present phantom and patient results. In this approach, multi-bed clinical oncology scans can be ready for diagnosis within minutes.