How exactly are you streaming it?
I would imagine that the biggest bottleneck would be your home network, rather than your computer hardware. Is it wireless, or is it connected up via ethernet?
Also, do you mean that you wish to stream simultaneously to 3 consoles? It's very difficult to say what exactly you need (and I have no experience in streaming 3 feeds all at once), but I think it might be fair to suggest at least a decent quad-core processor and at least 4GB of RAM.
Also, I would imagine that the video card installed might have the biggest impact. I think that most of today's GPU's have video acceleration built in to encode the images. I have no real idea of the minimum specs you should be looking for, but to be on the safe side I'd suggest looking at recent mid-range video cards.
Don't make any decision until someone else can contribute to this answer though, as all I've given you is an educated guess.