We've checked our permissions, and our users have read access to the subsites in question. The site collection admins can view all pages just fine. Normal users can access the content by accessing the site through http://[server]/sites/Test/Subsite/_layouts/sitemanager.aspx. The problem only appears on subsites. People just get "Access Denied" error messages unless they have SCA access.
We think it might be related to the default.master, but it seems ok. The default.master permissions are correct, it's not in draft mode, and we've opened it matches working default.master pages on other sites.