From what I can tell, all of those borders are there because of this css:
border: 1px solid #AAAAAA;
This css can be found in the jquery-ui.css file within your skin. You could open it and find .ui-widgit-content and then just delete the entire border rule, that should fix it.
That will, however, remove the border from legitimate jQuery UI items as well, so you might be better off trying to figure out why the ui-widgit-content class is being added to so many divs. My guess is that this is somewhere in your theme.skin, so you might look in there for it and see.
Hope this helps,