Intersite interactions play a crucial role in polar gases in optical lattices even in the absence of hopping. We show that due to these long-range interactions a destabilized stack of quasi-one-dimensional Bose-Einstein condensates develops a correlated modulational instability in the nonoverlapping sites. Interestingly, this density pattern may evolve spontaneously into soliton filaments or into a checkerboard soliton crystal, offering a unique possibility to observe such two-dimensional arrangements in ultracold gases. These self-assembled structures may be observed under realistic conditions within current experimental feasibilities.