I just came across this too after flashing the stock rooted 4.3 ROM.
There seems to be no solution for this, and GS4 people have been complaining about it for a while. Haven't seen a real solution yet unfortunately.
In my case, my phone is on vibrate most of the time. When it's not on vibrate it can be nice to hear an audible notification when I receive a new email. But to be honest, audible Gmail notifications aren't that critical for me. So I set the Gmail notification ringtone to Silent and now my GS3 doesn't vibrate when I receive emails.