The answer turns out to vary depending on what material you're talking about.
In metals and in typical semiconductors, the electrons are present as spread-out quantum waves. These accelerate in response to an applied field, then occasionally scatter off an impurity or a little sound wave, then re-accelerate, etc, so they have some average velocity in the direction the field is pushing them.
In some types of semi-conducting material, the electron waves spend most of their time locally trapped at individual atoms. They occasionally hop to another atom. When a field is applied, there are a few more hops in the direction the field is pushing than in other directions, giving an average current flow.
These aren't the only cases, but they give some feel for the variety.
(published on 07/30/2009)