The selection of the correct model for describing contaminant transport in fractured media remains a challenging problem in subsurface hydrology. In this work, the experimental data corresponding to nonsorbing tracer runs in natural fissures of Moreno et al. (1985) are analyzed. The advection dispersion equation (ADE) model yields a poor fit in the late time regions of the experimental breakthrough curves denoting the existence of anomalous or non-Fickian transport. Continuous time random walks (CTRW) are proposed for describing this kind of transport using two of the most studied forms for the transition rate probability: the asymptotic and truncated power law models. The CTRW approach produces better fits to the experimental data than the ADE model. In all the simulations a moderately dispersive transport was found which is interpreted as being as a result of the effects of the small- and intermediate-scale heterogeneities.