Yup, you can use Nvidia video cards on AMD boards and ATI cards on Nvidia based motherboards. It makes absolutely no difference in the performance of the video cards.
Now in regards to Crossfire and SLI, Crossfire is only supported on AMD and some Intel boards using ATI video cards. SLI is only supported with Nvidia video cards on Nvidia based motherboards and on some Intel X58 motherboards.
The HD 4870 is a nice deal on a very good card. It should work well for you.