I have told people for years to not import "bare" MTS files (ie copied outside of the AVCHD package) because it can create significant I/O performance problems. FCP was evidently trying to dynamically rewrap each clip upon each reference.
That said, importing "bare" MTS files has nonetheless worked for 10 years. MTS files are mentioned as supported on Apple's "Media Formats Supported" page: Media formats supported in Final Cut Pro for Mac - Apple Support
The MTS audio problem with 10.8 is not limited to newly-imported files. Any prior version library containing "bare" MTS files which is updated to 10.8 will stop playing audio from those files. This is regardless of whether the files were imported using "leave files in place" or "copy to library."
In many cases the files were imported years ago and the customers no longer have the original AVCHD bundle.
I tried using EditReady to rewrap MTS files before importing to FCP 10.8 and that seemed to work. That might be a solution for newly-imported material.
However for existing MTS material in libraries and projects, the solution is less clear. I tried relinking to a few rewrapped files. Some worked and some failed due to the error "The original file and new file have no
shared media range." It will take more investigative work to determine what the options are for customers in this situation.
This should probably be filed as a bug, so Apple can state whether the change is expected behavior as of 10.8 or a bug.
There is a similar format with file extension M2TS. I don't know if those are also affected.