Contrary to popular belief and Twitter’s terms of service, you cannot copyright a Tweet. Under US law, copyright is granted on publication to “original works of authorship” finalized in “fixed forms of expression” but this does not extend to names, titles, or short phrases (PDF).
As messages sent via Twitter cannot be longer than 140 characters, they cannot be copyrighted. However original, witty, or profound they may be, nothing more than good manners protects your original expression of authorship. If you wish to let other people quote or use your Tweets, you need not “license” them; indeed, technically, you cannot license them, since they are in the public domain the instant you publish them.
If you write a clever Tweet and wish to assert ownership (and if money is no object), you may apply for a trademark. Good luck with that.
Otherwise, your Tweets are like the air. Anyone can do anything like to them, including quoting them with or without your permission. If an enterprising company wants to take something you said on Twitter and slap it on a tee shirt, they may do so. If a gent of the disturbed persuasion wants to engrave your tweet into a 600-foot swastika, he may do so.
If this disturbs you, suck it up, or stop using Twitter—or mark your Twitter feed as private. This will not copyright your Twitter mutterings but it will keep many people from seeing them.
If it deeply disturbs you (and money is no object), mount a case to change the law.
Me, I plan to use Twitter forever. And any party so inclined may make a whistle of my Tweets. But my saying so here is irrelevant because you cannot copyright a Tweet.
Update: Comments are now closed, but you may read what others had to say. Thanks to all for a lively and illuminating discussion.