The interaction ('cross terms') between diffusion-weighting gradients and susceptibility-induced background gradient fields around vessels has an impact on apparent diffusion coefficient (ADC) measurements and diffusion-weighted functional magnetic resonance imaging (DFMRI) experiments. Monte-Carlo (MC) simulations numerically integrating the Bloch equations for a large number of random walks in a vascular model were used to investigate to what extent such interactions would influence the extravascular signal change as well as the ADC change observed in DFMRI experiments. The vascular model consists of a set of independent, randomly oriented, infinite cylinders whose internal magnetic susceptibility varies as the state changes between rest and activation. In such a network, the cross terms result in the observation of a functional increase in ADC accompanied by a descending percent signal change with increasing diffusion weighting. It is shown that the twice-refocused spin-echo sequence permits sufficient yet not total suppression of such effects compared to the standard Stejskal-Tanner spin-echo diffusion weighting under experimentally relevant conditions.