Context. Analyses of Planck data have demonstrated that the grain alignment efficiency is almost constant in the diffuse and translucent interstellar medium (ISM).Aims. We aim to test whether the radiative torque (RAT) theory is compatible with these new observational constraints on grain alignment.Methods. We combine a numerical magnetohydrodynamical simulation with a state-of-the-art radiative transfer post-processing code POLARIS which incorporates a physical dust model and the detailed physics of grain alignment by RATs. A dust model based on two distinct power-law-sized distributions of spherical graphite grains and oblate silicate grains was designed to reproduce the mean spectral dependence of extinction and polarization observed in the diffuse ISM. From a simulation of interstellar turbulence obtained with the adaptive-mesh-refinement code RAMSES, we extracted a data cube with physical conditions representative of the diffuse ISM. We post-process the RAMSES cube with POLARIS to compute the grain temperature and alignment efficiency in each cell of the cube. Finally, we simulate synthetic dust emission and polarization observations.Results. In our simulation, the grain alignment efficiency is well-correlated with the gas pressure, but not with the radiative torque intensity. Because of the low dust extinction in our simulation, the magnitude of the radiative torque varies little, decreasing only for column densities larger than 1022 cm−2. In comparing our synthetic maps with those obtained assuming a uniform alignment efficiency, we find no systematic difference and very small random differences. The dependencies of the polarization fraction p with the column density NH or with the dispersion in polarization angle S are also similar in both cases. The drop of grain alignment produced by the RAT model in the denser cells of the data cube does not significantly affect the patterns of the synthetic polarization maps, the polarization signal being dominated by the line-of-sight and beam integration of the geometry of the magnetic field. If a star is artificially inserted at the center of the simulation, the polarization fraction is increased everywhere, with no specific pattern around the star. The angle-dependence of the RAT efficiency is not observed in simulated maps and where the magnetic field is artificially set to a uniform configuration in the plane of the sky, it is only seen to be very weak in the optimal configuration.Conclusions. The RAT alignment theory is found to be compatible with the Planck polarization data for the diffuse and translucent ISM in the sense that both uniform alignment and RAT alignment lead to very similar simulated maps. To further test the predictions of the RAT theory in an environment where an important drop of grain alignment is expected, high-resolution polarization observations of dense regions must be confronted with numerical simulations sampling high-column densities (NH > 1022 cm−2) through dense clouds, given a sufficient statistical basis.Key words: submillimeter: ISM / magnetic fields / radiative transfer / polarization / dust, extinction