Fortunately for all of us, Blizzard has announced that instead of waiting on hold, you can now fill out a form online detailing the problem you're having and they'll give you a call. This means no more phone menus or explaining your problem to a support agent -- they'll have your account info and problem info at hand when they call, so you can dive right into troubleshooting your problem. The feature is available now between the hours of 1PM and 9PM EST (or 10AM and 6PM PST) -- off-hours, you'll just have to make a support ticket, like always, and wait it out. But in case your problem turns out to be an easy fix, the site will also suggest some support documents to help you out.
To use the new system, just head to Battle.net's support page and select Get Help to see your contact options.