Ben's partially correct. Unfortunately Google has been claiming they do process Javascript for a while, and they recently stated they've begun reading AJAX. Of course they do a lousy job of it and don't always get it right, which just makes things even more muddy.
So from an SEO best practices perspective, you shouldn't have the menu(s) in the first place, at all.
You may also THINK their good for users but has any significant study been performed to confirm that? You'd need to check click-through rates on all the links to know for sure.
What I've found through years of auditing sites that have such menus is that it almost always turns out to be the case where most of the deeper links NEVER get clicked on from within these menus. Instead, they're overwhelming to users. This is why it's better to not have them from a UX perspective.
If you abandon them and go with more traditional hierarchical index and sub-index pages, and if those are properly optimized, you'll not only eliminate the massive SEO problem but in fact get more of your category pages to have higher ranking strength and authority over time.
IF you're going to keep them in any form because you don't want to go to the extreme I recommend, then yes - AJAX would likely be the only scenario that offers the least likelihood of search engines choking on the over-use of links.
And for the record, the real current problem with all those links on every page is duplicate content confusion - all of those URLS at the source level dilutes the uniqueness of content on every page of the site. And that also means you're harming the topical focus of every page as well. So whatever you do, AJAX or doing away with them altogether is going to be of high value long term.
- Alan Bleiweiss - Click2Rank's Search Team Director