i known it's old, but i found this when searching for a solution. Meanwhile, i think i found a solution. In the email app, use phone's Menu button (lower left on S3), click settings. There's a "General preferences" at the top, followed by the list of "Added accounts". I had changed the ringtone in the General preferences before, and it had no effect. But now, I went into each account, near the bottom there's a "Select ringtone" from which I could change it to None. This seems to have worked for me. Note that the "Select ringtone" is greyed out if the phone volume is off.
galaxy s3, virgin mobile canada, android 4.1.2.
