Cosmological models based on f(R)-gravity may exhibit a natural acceleration mechanism without introducing a dark energy component. We investigate cosmological consequences of the so-called Hu-Sawicki f(R) gravity in the Palatini formalism. We derived theoretical constraints on the model parameters and performed a statistical analysis to determine the parametric space allowed by current observational data. We find that this class of models is indistinguishable from the standard ΛCDM model at the background level. In contrast to previous results achieved with the metric approach, we show that these scenarios are able to produce the sequence of radiation-dominated, matter-dominated, and accelerating periods without need of dark energy.