The first thing I would check is that site assets like the logo and the master page have all been checked in, published, and approved.
The next question I would ask is do you have any custom components on the page trying to be accessed? If yes there may be pieces of code trying to perform operations that users with restricted read are not able to do. This would cause an Access Denied error. You can also check this by seeing if the user can access an application page such as the View All Site content link because there should be no custom components there. (http://yoursite/subsite/_layouts/viewlsts.aspx
Of course, check the server event and ULS logs in hopes of more details. Your best bet would be to get on the phone with a user having troubles (or just use an account that can reproduce the issue if you have one available) and immediately after generating an Access Denied message open up the ULS log and what you are looking for should hopefully be near the bottom of the log file.