SOLVED! At least for me.
The trick is to turn off (or otherwise customize) notification for the specific account:
Email > Settings > {account name} > Email Notifications = OFF
or
Email > Settings > {account name} > Select Ringtone > Silent
With the latter option, you'll still get the icon in...
FWIW, I'm having the same problem: Credo/Sprint Samsung Galaxy SIII, JB 4.1.1
Furthermore, when I leave email notification ON, but change the ringtone to something else or Silent, I still hear the original ringtone (Postman) when email arrives.