We make a systematic study of the cosmological dynamics for a number of f(R) gravity theories in Palatini formalism, using phase space analysis as well as numerical simulations. Considering homogeneous and isotropic models, we find a number of interesting results: (i) models based on theories of the type (a) f(R)=R-beta/Rⁿ and (b) f(R)=R+alphalnf R-beta, unlike the metric formalism, are capable of producing the sequence of radiation-dominated, matter-dominated, and de Sitter periods, and (ii) models based on theories of the type (c) f(R)=R+alphaR^m-beta/Rⁿ can produce early as well as late accelerating phases but an early inflationary epoch does not seem to be compatible with the presence of a subsequent radiation-dominated era. Thus, for the classes of models considered here, we have been unable to find the sequence of all four dynamical epochs required to account for the complete cosmological dynamics, even though three out of four phases are possible. We also place observational constraints on these models using the recently released supernovae data by the Supernova Legacy Survey as well as the baryon acoustic oscillation peak in the Sloan Digital Sky Survey luminous red galaxy sample and the cosmic microwave background shift parameter. The best-fit values are found to be n=0.027, beta=4.63 for the models based on (a) and alpha=0.11, beta=4.62 for the models based on (b), neither of which are significantly preferred over the LambdaCDM model. Moreover, the logarithmic term alone is not capable of explaining the late acceleration. The models based on (c) are also consistent with the data with suitable choices of their parameters. We also find that some of the models for which the radiation-dominated epoch is absent prior to the matter-dominated era also fit the data. The reason for this apparent contradiction is that the combination of the data considered here does not place stringent enough constraints on the cosmological evolution prior to the decoupling epoch, which highlights the importance of our combined theoretical-observational approach to constrain models.