I like Twitter (though I don't know how I would use it in teaching).
I also have a short attention span.
Same here. I think it's much more effective when people are using portable web devices--it rarely occurs to me to send a tweet when I'm just sitting in front of my computer. But Twitter can be useful for things like liveblogging events (although there was a dust-up after a recent academic conference, where apparently the twitterers were less than enthusiastic about a keynote speech, and offended a lot of their followers with their unprofessional comments).
In my field, Twitter is starting to have some interesting applications. I agree, though, that I'm not sure how I could fit it into my teaching, or why I'd want to.