Galaxy clusters, the most massive collapsed structures, have been routinely used to determine cosmological parameters. When using clusters for cosmology, the crucial assumption is that they are relaxed. However, subarcminute resolution Sunyaev-Zel'dovich (SZ) effect images compared with high-resolution X-ray images of some clusters show significant offsets between the two peaks. We have carried out self-consistent N-body/hydrodynamical simulations of merging galaxy clusters using FLASH to study these offsets quantitatively. We have found that significant displacements result between the SZ and X-ray peaks for large relative velocities for all masses used in our simulations as long as the impact parameters were about 100-250 kpc. Our results suggest that the SZ peak coincides with the peak in the pressure times the line-of-sight characteristic length and not the pressure maximum (as it would for clusters in equilibrium). The peak in the X-ray emission, as expected, coincides with the density maximum of the main cluster. As a consequence, the morphology of the SZ signal, and therefore the offset between the SZ and X-ray peaks, change with viewing angle. As an application, we compare the morphologies of our simulated images to observed SZ and X-ray images and mass surface densities derived from weak-lensing observations of the merging galaxy cluster CL0152-1357, we find that a large relative velocity of 4800 km s-1 is necessary to explain the observations. We conclude that an analysis of the morphologies of multi-frequency observations of merging clusters can be used to put meaningful constraints on the initial parameters of the progenitors.