All cards come with reasonable cooling solutions. Some are better than others. It is commonly accepted (in your price range) that the 570gtx has the best stock cooler. Most tend to exhaust their heat through the back of the case but still, be prepared for increases in case temperature. Are all of your fan slots taken up?
Another few issues:
a) If PSU is only 300W, it will need upgraded. Post the 12V amp rating (sometimes on the back/sides of the PSU)
b) i5 2.67 will likely bottleneck the a 570gtx and 6970 (perhaps maybe even a 6950)
c) What resolution do you game at? You will need at least 1080p or again video card becomes bottle necked because of low resolution and low cpu power.
d) How long a card will your case accept? (570gtx/ 6950 and 6970 are around 270mm in length. Next batch of cards are around 250mm, then the next 220mm. Googling the gt220 length suggest 160-170mm for comparison)