And no, I was not just thinking of ionic solids. For instance take ice. A molecule of H2O has no net charge, but locally there is a pattern of positive and negative charges on the surface which results in a predictable crystal structure. Most molecules aren't that strongly polarized, but no matter what there is always some pattern of charges, and atoms will tend to line up so that positive meets negative and they attract. They will do this even when passing by quickly, hence the Van der Waals forces.
Consider for instance Sodium. If there were no force other than Van der Waals holding the atoms together in the crystal, there is no explanation for the fact that Argon, a significantly heavier atom, has much lower melting and boiling points.
One last comment then:
And what causes that crystalline microstructure other than the pattern of positive and negative charges on the atoms involved, resulting in adjacent atoms being attracted to each other?
You must be thinking of an ionic solid such as NaCl, but there are many solids (iron and silicon come to mind) in which the atoms have no net charge, yet the material is a crystalline solid. My point is that there is more at work here than just electromagnetic forces, namely quantum mechanics. Similarly, refering to our original discussion, there is more than just electrostatics that is responsible for the hardness of crystalline solids.