So, having gone through all the links shown in GWT and removed those that I can that I don't believe are beneficial, I go back to GWT and they are still there. Weeks and weeks go by, and they won't go away. My understanding is that unless the sites where the links are coming from are updated, they won't get recrawled. I haven't had an unnatural links warning, so I can't notify Google that these links are gone via a reconsideration request.
Google encourages webmasters to look at where there links are coming from, but there doesn't seem to be any machinery in place to let them know the work that has been done. Unless I am missing something?