  We introduce a Monte Carlo algorithm to efficiently compute transport
properties of chaotic dynamical systems. Our method exploits the importance
sampling technique that favors trajectories in the tail of the distribution of
displacements, where deviations from a diffusive process are most prominent. We
search for initial conditions using a proposal that correlates states in the
Markov chain constructed via a Metropolis-Hastings algorithm. We show that our
method outperforms the direct sampling method and also Metropolis-Hastings
methods with alternative proposals. We test our general method through
numerical simulations in 1D (box-map) and 2D (Lorentz gas) systems.
