How to sync bookmarks between Firefox, Chrome, and Safari on multiple computers?

Xmarks (formerly FoxMarks) could solve your problem. It can synchronize (privately) bookmarks from Firefox, Safari, and Internet Explorer.

EDIT: Lifehacker has an update on Chrome supporting bookmark sync

EDIT: As of May 1st 2018, Xmarks has been shut down by its parent company, LastPass.

Chrome does have an extension for Xmarks (formerly FoxMarks) now. I just installed it today, and it works beautifully. I had already had an account set up through both Internet Explorer and Firefox, but I recently had switched to Chrome. I was not looking forward to having to find another online synchronization program.

For 2012, XMarks came back from the grave as part of LastPass. It supports 4 of 5 major browsers (IE, Firefox, Chrome, Safari) but lacks Opera. It's not clear whether this is due to Opera API issues, but there is a little discussion in their forums. Looks like it's free, but you have to subscribe for use with mobile devices/tablets (iDevice, Android, etc.) and a few other features. This is the same strategy used by LastPass. (Disclaimer: premium user of both)

Google Bookmarks may be an option depending on your browser choice and plugin availability. Google Chrome Sync (and possibly plugins for other browsers) may also work; for Firefox I believe it's tied in with the Google Toolbar for Firefox but YMMV by browser.

Delicious may be an option; I used to be a user but have mostly stopped after the headaches with their switchover (I switched to XMarks). I think they were trying to go in a more "social" direction like Digg, et al.

Pinboard is a commercial service (one-time fee) that apparently also uses the Delicious API, so some addons/widgets may be able to work with it instead of with Delicious. Long-term support of those may be questionable if Delicious doesn't still support that API.

UPDATE: For XMarks, I'm not sure what their plans are for IE10, but I'd expect them to only support the Desktop version of IE - there's apparently also a "Metro" version that will not allow any plugins.

It is now 2012, and Xmarks is still the best solution.
Bought by LastPass, its existence and continuing free status are not any more in doubt.

It now fully supports Firefox 3+, Internet Explorer 7-9, Safari 4-5 and Chrome, all available from its download page.