We examine closely the solar center-to-limb variation of continua and lines and compare observations with predictions from both a three-dimensional (3D) hydrodynamic simulation of the solar surface (provided by M. Asplund and collaborators) and one-dimensional (1D) model atmospheres. Intensities from the 3D time series are derived by means of the new synthesis code ASSɛT, which overcomes limitations of previously available codes by including a consistent treatment of scattering and allowing for arbitrarily complex line and continuum opacities. In the continuum, we find very similar discrepancies between synthesis and observation for both types of model atmospheres. This is in contrast to previous studies that used a ``horizontal'' and time-averaged representation of the 3D model and found a significantly larger disagreement with observations. The presence of temperature and velocity fields in the 3D simulation provides a significant advantage when it comes to reproducing solar spectral line shapes. Nonetheless, a comparison of observed and synthetic equivalent widths reveals that the 3D model also predicts more uniform abundances as a function of position angle on the disk. We conclude that the 3D simulation provides not only a more realistic description of the gas dynamics, but despite its simplified treatment of the radiation transport, it also predicts reasonably well the observed center-to-limb variation, which is indicative of a thermal structure free from significant systematic errors.