For almost two years I've used a single 6970 2gb with three 1600x1200 monitors in eyefinity. The 6970 is similar in performance to the 7850 2gb. At this level of GPU performance, high resolutions are very possible, but demanding games require reduced settings to get playable framerates.
If you require ultra settings then you'll want to get something more powerful. If you are willing to accept reductions in game settings it'll work fine. Likewise, if you are the type of gamer that requires 60 fps at all times, you will want to get something more powerful.
With each game I usually spend about a hour tweaking individual settings and in today's games can get medium to high settings with a little MSAA and no less than 40fps (I'm not very picky but anything less is noticeably less-smooth to me). My point is that if you are the kind of gamer that is willing to sacrifice game settings for some awesome peripheral views, you can tweak the settings down or up to get smooth-enough play.
Another thing to remember is that if a particular game isn't running well-enough (or doesn't work on three screens) then you simply play it on your single center screen like you're doing right now - just set the in-game resolution to 1600x900.
One more thing - while the 7850 2gb is a good card, you will find that you pretty much want as much as you can get when you go triple monitor resolution. I'd love to just max out games, but alas, I'm not a millionaire. For the past year I've salivated over a 7970 or a 680/670 but decided to wait it out. From what I've read, it looks like a long wait for the next gen cards though. If you can squeeze out a 7950 or 7970 you'd be happier with the performance methinks. The 7850 2gb and my 6970 do great, but I think you should pick up a 7970, then save up for the extra two monitors and rock out the triple screens then.