I’ve looked online: some folks say yes, then people can find you. Some say no, putting your name out there is bad. I figure I’m never going to be the internet mystery celebrity, so why not?
Probably. I do; it helps me remember that I’m not anonymous and I need to be aware of the things I say. I use a pseudonym here because it’s the style, but most places on the Internet I’m just me.
If you plan to express controversial opinions, then using a pseudonym may be a good idea - but make sure you take care to separate your real identity from your online one. Only browse in Internet cafes or libraries, never access your Twitter account from a machine that has also accessed your personal Facebook or Gmail accounts, that sort of thing.
Or just suck it up and use your name. 
Don’t use your name unless you’re planning on making yourself famous, or are already a politician/celebrity and wish to communicate with your constituents/fans. Nothing goes away on the internet, and who knows when you be interviewing for a job, or in another situation where someone has some power to make your life better or worse? They Google you, and they find you’ve posted an opinion they don’t like. Bam, you’re done. Hell, people get fired for voting for the wrong person, so I figure the less I put out there, the better.
This is exactly why I do use my real name. It helps me remember that what I put out there is going to be attributed back to me. I don’t have a dangerous belief in anonymity protecting me.
If you’re going to be expressing opinions so controversial that you’d be in danger of suffering serious consequences, then either don’t do so, or take steps to protect your identity - as I noted above, be serious about it.
If, on the other hand, you’re just being nervous about getting fired because you like Obama, or don’t like Obama, or whatever, then find a better job. One that doesn’t involve working for jerks who fire people for having opinions.
I’d advise against it and here’s why… on another board I’m on, the only way you can post is through Facebook, which unfortunately means you are supposed to use your real name. Some of the discussions on there can get pretty heated and, in usually rare circumstances, someone who is getting their butt resoundingly whipped, will look up another user and then post their personal details in an attempt to use that against them.
In itself, this isn’t that big a deal if your security settings are high enough. However, if your real name is fairly unique, having that piece of information is now only a couple of keystrokes away on google to find out any and everything they’d like to know about you.
That could be problematic if they send a derogatory note to your supervisor or gank your online pictures and post them elsewhere without your permission. The truly bad and scary part though, is anyone who dislikes you having your address at their fingertips. Not an issue if your detractor lives half a world away, a huge problem if they live one town over.
The last thing anyone wants is some random pissed off internet psycho boiling your bunny, leering at your kids while they’re at recess or harassing your spouse during a shopping excursion. I mean, the least of it is someone finding this stuff out about you and subscribing you to NAMBLA. The worst is what just happened two days ago in my former town of Tyler, Texas; a man killed over a tweet.
Just something to consider.
It depends on what you want to say. I don’t post anything on twitter that I would be embarassed to have read by a potential employer, so I use my real name. If you want to post controversial/inflammatory rhetoric, I recommend anonymity.
As a side note: all tweets are now recorded in perpetuity by the Library of Congress in a searchable database. You may want to consider the tone or content of your tweets, regardless of whether you use your real name or not.
Different types of social media are used differently – some, like both Facebook and Twitter are used for actual people interacting, though both of those have corporate [in a broad sense] interests jumping on the bandwagon as well.
I’m a freelancer, so I use Twitter for promotion and networking, not just shooting off at the mouth. My username is first name/last initial/subject tag, but I also provide my full real name and my city on Twitter, and provide my handle both in my email sig and on my business card. Thus I don’t say anything I wouldn’t want a potential client to see. (Which doesn’t mean everything is all business all the time – and god knows I tweet on politics a fair amount – just that I don’t act like an asshole. Which I try not to online in general. Or in real life, for that matter.)
A message board is a different kind of online animal, and comparing how you use the SDMB to how you use Twitter completely misses the point of both.
And just about everything posted on the internet is stored in the greatest repository of them all, Google Search.
I think I’ll compromise and do first name last initial.
Obligatory link to ominous-sounding government exploitation of freely accessible data:
It depends on your intended audience. It’s not going to do any good to hide your name if you’re going to be handing out business cards with your Twitter on it, for example. If the point of your Twitter is to interact with people who know who you are, use your real name. If it’s to interact with the Internet at large, don’t. And, yes, you can definitely be freer in an anonymous account.
Personally, I have no business reason to use Twitter, and it’s often an alternative to logging in with Facebook on places that require you to log in via other sites. So I do not use my real name. Though I do use the same fake name in a lot of places (not here), so I do wonder about that.