We present results from three-dimensional numerical simulations of quasi-perpendicular collisionless shocks in order to determine whether cross-field diffusion, which is otherwise artificially suppressed in simulations which contain at least one ignorable spatial coordinate, is large enough to efficiently accelerate thermal particles and/or pickup ions. We find that although the simulated downstream distribution functions are quite steep, a fraction of the particles are accelerated to energies well above the thermal energy. This occurs only if the system contains fluctuations with wavelengths that are considerably larger than the gyroradii of the particles of interest (the high-energy ones). We find that this is due to the fact that the transport of the particles normal to the shock, against the downstream convection, is mostly influenced by the meandering of field lines on large scales. We also show that if the system does not contain these long-wavelength waves, the scattering is not sufficient to accelerate thermal particles or pickup ions. We use two different types of simulations to demonstrate this. In the first set of simulations we use the well-known hybrid simulation which treats the interaction between the particles and fields self-consistently. For computational tractability these simulations use very small spatial domains, and the effect of the large-scale field-line random walk is suppressed. However, this approximation accurately addresses the physics of the scattering at resonant wavelengths. Test-particle simulations, which are more computationally tractable than the hybrid simulations, are also performed for larger systems and to illustrate the effect of the long-wavelength waves. ¿ 2000 American Geophysical Union