You need to show your passport before you leave the country. If you are flying American, you should enter your passport number online along with contact information etc. I don't know if this is true for all airlines.
Sometimes there is a very short line in GC but if a lot of planes come in at the same time, it might be a bit of a wait.
I had heard that they were making some changes leaving Cayman, but don't know if that has happened yet.
Coming back into the US, we fly through Miami, and sometimes there are long lines thru security, and waits for luggage, but generally the passport checks are very quick. (And sometimes, it's fast......I'll hope for fast for you)
Edited: 3:11 pm, August 25, 2012