You would have to use flexible PCIe-extenders to connect four GPUs because if you plug them all directly into MB, they'll block ait intakes of each other.
And you can't plug them all directly because standart ATX system would accept only 7 cards (and each 5870 takes double space).
For 4 cards you would need 3 spaces in between, so a total of 7 slots would suffice. Of course, for cooling you need extenders. 1x extenders are probably better for airflow and extra flexibility, and the 1x speed is not a concern for mining.
There are some Lian Li cases out there that have 8 card slots in the back. I'll have to agree, though, that sandwich of 4 5970s would be crazy hot. You'd have to take the case door off and stick a small box fan beside the computer and even then they would be a little hot.
Also, there are motherboards where the 7th slot is an x16 slot, which the Lian Li case would work with. Front panel wiring be damned.