This paper questions the validity of modulation theories that ignore the presence of large regions of enhanced magnetic field in the heliosphere. We simulate the scatter-free and diffusive motion of relativistic cosmic rays in models of the heliosphere magnetic field which include compressed field structures representing the corotating interaction regions which are believed to be present in the interplanetary medium throughout most of the solar cycle. We find that the strong field gradients associated with these regions greatly perturb the drift pattern anticipated for simple Parkerian fields, and thus the expected streaming of cosmic rays from over the poles or along the current sheet, during consecutive solar cycles, no longer holds.