I've been looking at this problem again after sidelining it for a couple of months as I was busy elsewhere. Thanks, everyone, for your responses so far but I am still having the same problem...
I installed a new theme to see if the same problem would occur, and it did. With a little more investigating, I think the problem lies with PHP includes in my text widgets. I have a large number of links in various different categories and don't want them all displayed in one dropdown list, so in a couple of places on my sidebars I used text widgets with PHP includes inside them, in order to display lists of links from a specific category only, ie:
<?php wp_list_bookmarks('title_li=&categorize=0&category=24&show_description=1&after=</br>&orderby=url'); ?>
For whatever reason, IE doesn't display these lists properly and it's this that messes up the layout. However, when you use a text widget containing just regular text, without the PHP include, it works fine.
Hope this makes sense! I still don't know how I can get around this and get IE to display the links in my widgets properly, but I thought pinpointing the source a little more might help someone answer.