This is how I did it. It has nothing to do with WordPress, but could work in any kind of site. It uses Javascript.
http://webdesign.about.com/od/sound/a/play_sound_oncl.htm
I think you could do it with JQuery, too.
The truth is, I hated this effect - it's not something I would ever do on my own site. But it got the job done when I was hired to do it.
There's no WP plugin for this method that I'm aware of. You'd need to find a way to get this onto links, paragraphs, what-have-you. It was not hard to do in MODx - I'm not sure how I'd do it in WP. You might need to hard-code your menu (yuck!) unless you can figure out a way to insert the javascript onto links or wherever. Oh, you could just stick the code into a widget using the HTML widget, and that would work.
If you do this, and I don't recommend that you do, use VERY small sound files. Otherwise your site usability will be very bad, along with having possibly annoying sounds.
Good luck