Semiconductors and insulators owe their low conductivity to the properties of the valence band in those materials. It just so happens that the number of electrons is precisely equal to the number of states available up to the top of the valence band. There are no available states in the bandgap. This means that when an electric field is applied, the electrons cannot increase their energy (i.e. accelerate) because there are no states available to the electrons where they would be moving faster than they are already going.
There is some conductivity in insulators, however. This is due to thermal excitation - some of the electrons get enough energy to jump the bandgap in one go. Once they are in the conduction band, they can conduct electricity, as can the hole they left behind in the valence band. The hole is an empty state which allows electrons in the valence band some degree of freedom.
It is a common misconception to refer to electrons in insulators as "bound" - as if they were somehow attached to the nucleus and couldn't move. Electrons in insulators are quite free to move - in fact they move at a speed on the order of 100 km (60 mi) per second! They are also delocalised, having no well defined position within the sample. It's just that for every electron moving left, there is another moving right - all the speeds cancel out precisely, leaving no overall current.