If its possible to fix the site without altering the directory and URL structure, that's going to be your best option.
However, I'm guessing you're going to be fixing URLs as well, and if that's the case you'll want to use 301 redirects to tell search engines where the content has gone. See http://www.tamingthebeast.net/articles3/spiders-301-redirect.htm
Finally, wherever possible try to get the owners of the sites with backlinks to update their links to you.