Is it possible to configure the new Mac Pro hard drives so that bay 1 and bay 2 are one, and bay 3 and bay 4 are another, no mirroring, just two separate hard drives? And if so, can I do it without buying that RAID card?
This should be possible. You would need to reinstall Mac OS X. When you do you would go into Disk Utility and create a RAID disk. In your case two RAID Disks. Then you would just set Disk 1 and 2 in RAID 0 (Striped) and Disk 3 and 4 in RAID 0.
However I don't think these stripped arrays would work with bootcamp. So if you ran Windows you would need to leave some disk space for Windows to use. Though I may be wrong about this last point.