Not true. I saw a Nova or Discovery Channel special on this. The knee-jerk reaction of quickly grabbing your hand away from a hot griddle is NOT something run by your brain. Pain such as this only makes it as far as your spinal cord before forcing you into action. The pain message cannot go all the way to your brain, wait for your brain to make a decision, and then send the message to your hand that it needs to move... That instantaneous reaction to grab your hand away has to be automatic or you might get a bad enough burn that you actually lose the use of your hand. Only after you're already removing your hand from the burning griddle does your brain get the message "Damn, that's hot".
IMO, fish MUST feel pain. Think about it. Say a larger fish comes along and takes a bite at his tail. Yeah, he'll flinch away, but then he will also avoid that size/type of fish in the future, because he associates the sensation of pain with that fish. Natural selection would weed out any fish that experienced pain but survived only to get right back into the same situation all over again because it didn't have any lasting memory of the pain.
Granted, fish don't have much of a thought process, but they feel pain sure as you and I do. And it has to be more than just a automatic flinch response, or all fish would have been wiped out long ago.