  We make a systematic study of the cosmological dynamics for a number of f(R)
gravity theories in Palatini formalism. We find a number of interesting
results: (i) models based on theories of the type (a) f(R)=R-beta R^n and (b)
f(R)=R+alpha ln 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+alpha R^m -beta R^n
can produce early as well as late accelerating phases. However 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 (SNLS)
as well as the baryon acoustic oscillation peak and the CMB shift parameter.
The best-fit values are found to be n=0.027, alpha=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 LCDM model. The models based on (c) are also
consistent with the data with suitable choices of their parameters.
