The solar oxygen abundance has undergone a major downward revision in the last decade, the most noticeable one being the update including 3D hydrodynamical simulations to model the solar photosphere. Up to now, such an analysis has been carried out only by one group using one radiation-hydrodynamics code. We investigate the photospheric oxygen abundance considering lines from atomic transitions. We also consider the relationship between the solar model used and the resulting solar oxygen abundance, to understand whether the downward abundance revision is specifically related to 3D hydrodynamical effects. We perform a new determination of the solar photospheric oxygen abundance by analysing different high-resolution high signal-to-noise ratio atlases of the solar flux and disc-centre intensity making use of the latest generation of CO5BOLD 3D solar model atmospheres. We find 8.73 < logNoxygen/Nhydrogen +12 < 8.79. The lower and upper value represent extreme assumptions on the role of collisional excitation and ionisation by neutral hydrogen for the NLTE level populations of neutral oxygen. The error of our analysis is +-0.04 +- 0.03 dex, the last being related to NLTE corrections, the first error to any other effect. 3D ``granulation effects'' do not play a decisive role in lowering the oxygen abundance. Our recommended value, considering our present ignorance of the role of collisions with hydrogen atoms on the NLTE level populations of oxygen, is log(Noxygen/Nhydrogen) = 8.76 +- 0.07. The reasons which have led to lower O abundances in the past are identified as (1) the lower equivalent widths adopted, and (2) the choice of neglecting collisions with hydrogen atoms in the statistical equilibrium calculations for oxygen.