Keep in mind that everything about Google's algorithm is nothing more than informed speculation since the company keeps it a secret.
You may have changed how the site is structured and thus changed how the links point to your site. A link to www.example.com is not the same as a link to example.com unless you go through the extra steps of redirecting one of them. Same thing for a link that points to example.com/directory/page.htm when the redesigned page is now at example.com/new-directory/page.htm or what have you.
If you did change the URLs of any of the pages, be sure to redirect them using a 301 permanent redirect and Google will EVENTUALLY move most of the link credit to the new home. That process is not instantaneous, however, so even if you already did that, it might take a while to show up.
Although it wouldn't necessarily impact the incoming links, Google does re-evaluate the rank and authority of any page that changes significantly. It isn't hard to see why. A spammer could write a great, useful page about life insurance and then as soon as it ranks well change it to a ad-filled landing page. It is possible (although this is an UNINFORMED guess) that Google drops the links temporarily as part of this re-evaluation.
Google should sort it out eventually, but it might take a while, which isn't good new for most webmasters. It's a shame really, because it penalizes those who try and improve their websites and keep them up to date. Next time, try updating your site in sections. That way, you'll keep some of your historic value while the updated parts of the site are recovering their value. Then, when those have recovered, you can update another part.