From my own experience, the HD5970 will not naturally fit into the Antec Three Hundred Case, you'll have to buy a Haf X or something similar to accomodate this huge card.
After I had opened the packaging (now 7 months ago) I rushed to my computer to insert the card and i thought it would be a piece of cake and and when I realized there's no way it would fit I panicked. After a night's sleep I decided I won't be buying a new case cause I had just spend like $870 on a graphics card and that's enough. So I went to a tool shop and bought one of those tiny hand saw blades and me and my friends sawed the damn thing for like 3 hours and finally the card fit but just barely. The inner area is 1.25 inches or 3 cm too short =/. Proof of my experiences: