While you can play ME3 right off the bat, I wouldn't.
The experience is 1000000x greater if you know the story, experienced all the ups and downs that got Shepard to where he/she is.
I mean the opening sequence where Shep runs into Kaidan/Ashley...would mean nothing to you. But it's a cool moment for those who've played from the beginning. A simple example of how these moments are ultimately going to be lost on you because you don't know the story, the relationships and experiences that you not only go through as a player, but have to make impacting decisions. When characters are referenced as you play ME3, you'll be like...k whatever I don't care, where as a full playthrough player would be remembering why that character is dead, possibly because of their own choice, or why that character hates you/loves you, again, because of what you chose to do.
Why not play ME1 or ME2? Both could be acquired for real cheap I'm sure, both still hold up today as excellent games. Most guys in this thread have played them both just within the past few weeks...and it's damn fun everytime. So it's not as if you're playing some "old inferior/outdated game", because they aren't like that at all.