In Day Of The Dead, one of the scientists experimented on a Zombie-- "Bubba" or something, his name was-- and found evidence of consciousness. I'm not sure if this was ever followed up. I must catch up on the Romero movies.
The zombie's name was "Bub," and it's followed up with "Land of the Dead." "Diary of the Dead" and "Survival of the Dead" take place during "Night of the Living Dead" and "Dawn of the Dead," more or less, so they have less to do with zombie intelligence. And the most recent one is pretty terrible.