How exactly does the new Mountain Lion filesystem work?
Don't worry, the filesystem is really the same as in Lion. What you may have misunderstood is how the new iCloud integration works. Every app that supports it (it may even be built-in if you use the system-standard open/save components, I'm unsure) will display an iCloud tab in the Open/Save dialogs, but this is in addition to the standard Finder hierarchy that's been around forever.

This new iCloud storage doesn't allow nested folders, so you only get one level of hierarchy. The idea is to keep things simple à la iOS (it looks like and is conceptually the same as organizing apps on iOS).

For more detail and description, check out the iCloud and You section of John Siracusa's Mountain Lion review at Ars Technica.