I do not have direct experience with nVidia's 3d setups, so I may not be entirely accurate, but lemme get this straight...
You are considering three 24-inch monitors set up in portrait mode so you can game on an effective 3600 x 1920 screen? I will answer what I can regarding this specific question
First off, your GTX 570 can only support two monitors at a time and only one monitor for gaming at a time, so adding another GTX 570 SLI is an option. Doing this will allow you to do the nVidia Vision thing in 2D where you can get an effective resolution up to 4800x2560 (1600x2560 per display). I have read this is not a good gaming setup, as the GTX 570 lacks enough muscle for newer games. Also, a new GTX 570 is $200+, so I don't think this is a good option.
A single, high-end AMD Radeon 7950 (overclocked as much as possible) or even a 7970 (again, overclocked) would theoretically be an appropriate option for your needs. Either card supports what AMD calls Eyefinity, which will support up to 6 monitors at one time. In this case, I'd suggest starting with one card and see how that works for you. If gaming is not up to par, then a second in Crossfire would be an option. One decent Radon 7970 costs about $400, so this option ain't cheap. Also, it's a hot topic I'd rather not get into, but some people really dislike Radeon cards lately because of poor drivers, heat, micro-stuttering issues, Crossfire problems, etc. I would not recommend this option either.
The last option I can think of is selling your GTX 570 (or recycling it in a new build
) and purchasing a single new GTX 670 with 4 GB of RAM at about $380. This card will support three monitors in that nVidia Vision thingy, but gameplay will probably not be very good with newer games. If so, you may consider adding another GTX 670 for an SLI configuration. But still, gameplay may not be adequate. Realistically, I do not believe any present-day, practical setup of whatever GPUs will enable Battlefield 3 with 60 FPS at max settings at a resolution of 4800x2560. I could be wrong though.
Now, without knowing anything else about your situation, here is a suggestion I'd offer. Though it may not be what you are asking, why not consider one more monitor, say a 30-inch Dell IPS, instead of two additional 24-inchers? That way, you may keep the GTX 570 as-is and supporting both monitors, you would have decent gameplay on just the 30-inch panel, and you would have the extra space with the current 24-inch panel for general computing as well.