False. I keep seeing this come up, and it's normally followed by someone more knowledgeable saying that it's not true at all. I'm not sure how knowledgeable I am, but I know that any creture you can control in Fortress Mode can get strange moods. Mostly from hearing those knowledgeable people talk, mind you.
I think that it would take more than mere unhappiness to get a dwarf to leave the fort. Unless there's a dwarven community nearby, you're talking a several-week trek, on foot, with only as much in the way of supplies as you and any dwarves you can convince to go with you can carry, possibly through environments which are actively or inactively inhospitable to you, perhaps chancing attacks by goblins, kobolds, elves, or whatever, possibly having to beat back the hordes of the dead...the only way a dwarf might consider leaving the relative safety, with a huge chunk of earth over his head, one or two square meals a day, a place to sleep at night, alcohol, and (most importantly, from a psychological point of view) other dwarves is if the dangers at home outweigh the benifets of home and the dangers of leaving.
That said, it just occured to me that good biomes might cause dwarves to want to enjoy the pastoral wilderness and wander away. (Not neccesarily the outdoor wilderness, of course--good caverns get touched by good, too.)