  We examine the question as to whether the Palatini f(R) gravity theories
permit space-times in which the causality is violated. We show that every
perfect-fluid G\"{o}del-type solution of Palatini f(R) gravity with density
$\rho$ and pressure $p$ that satisfy the weak energy condition $\rho+p \geq 0$
is necessarily isometric to the G\"odel geometry, demonstrating therefore that
these theories present causal anomalies in the form of closed time-like curves.
This result extends a theorem on G\"{o}del-type models to the framework of
Palatini f(R) gravity theory. We concretely examine the G\"odel-type
perfect-fluid solutions in specific $f(R) = R - \alpha/R^{n}$ Palatini gravity
theory, where the free parameters $\alpha$ and $n$ have been recently
constrained by a diverse set of observational data. We show that for positive
matter density and for $\alpha$ and $n$ within the interval permitted by the
observational data, this theory does not admit the G\"odel geometry as a
perfect-fluid solution of its field equations. In this sense, this theory
remedies the causal pathology in the form of closed time-like curves which is
allowed in general relativity. We derive an expression for a critical radius
$r_c$ (beyond which the causality is violated) for an arbitrary Palatini f(R)
theory, thus making apparent that the violation of causality depends on the
form of f(R) and on the matter content components. We also examine the
violation of causality of G\"odel-type by considering a single scalar field as
the matter content. For this source we show that Palatini f(R) gravity gives
rise to a unique G\"odel-type solution with no violation of causality.
