The effect of finite Larmor radius on the transport of passive charged test particles moving in turbulent electrostatic fields is investigated. The turbulent field is governed by a flexible model which is able to produce turbulence where zonal flows are damped or free to self-generate. A subtle interplay between trapping in small scale vortices and entrainment in larger scale zonal flows determines the rate, character and Larmor radius dependence of the test particle transport. When zonal flows are damped, the transport is classically diffusive, with Gaussian statistics, and the rate of transport decreases with increasing Larmor radius. Once the Larmor radius is larger than the typical radius of the turbulent vortices, the rate of transport remains roughly constant. When zonal flows are allowed non-Gaussian statistics are observed. Radial transport (across the zones) is subdiffusive and decreases with the Larmor radius at a slower rate. Poloidal transport (along the zones), however, is superdiffusive and increases with small values of the Larmor radius.