I am in Kuwait on a tourist visa and my visa is in my maiden name. I have a new passport in my married name but I entered Kuwait with a tourist visa in my maiden name. Will I have any problems leaving Kuwait in June?
You'll be fine. Just bring both passports with you and show them at check-in and to immigration. Bringing a marriage/name change certificate won't hurt.
Note that I'm assuming your old passport is expired/cancelled now. If it's still valid, then you don't even need to show them the new one, although the airline may still want to see it due to passport validity rules at your destination.