Looks good to me. All it needs is to make sure everything "fits" which you never know until you get the parts.
(Asus) 7870 = PCI-E 3.0, your MoBO will have 1x 3.0, 1x 2.0 and plenty 1.0's.
i5 3570 = Socket LGA1155, your MoBo supports all i3/i5/i7 socket 1155's.
Ballistix = DDR3, 240 pin, 1600, your MoBo supports 240 pin DDR3 2400, 1600 and 1333.
Not sure how to check PSU compatability, that'll have to go to someone else.
Also, you're getting a x64 OS right?