Yes PCIe 3.0 is backwards compatible.
For $230 you should keep your eyes peeled for a Powercolor 7870 Myst. It features a Tahiti LE chip which means it's a crippled 7950 and at a $230 price point. http://www.newegg.ca/Product/Product.aspx?Item=N82E16814131484
you can also try to find the Sapphire version, the Sapphire 7870 XT. It is currently discontinued by NewEgg but it should just be a out of stock issue.