2x 5750 is a well known budget performance setup. It can compete very well with a $300 5850, and blow away a single 5770. Of those choices, that would be the one I would lean towards.
5570 and GT240 are not gaming cards, they are top tier HTPC cards, and I don't believe that the 240 even supports multi-card setups. 3870x2s are pretty old tech, and the power consumption alone would dissuade me from going that route.
Budget wise, you will pay a huge premium for triple PCI-E motherboards, with the cheapest running a x4 slot, which will cripple the performance potential of running 3 cards.
a third option would be to get a single PCI-E slot motherboard (save a few bucks) and max out the GPu budget on a single 460 768mb model, or a 5830. These cards can be found for less than 200 with a bit of dedication.