Platter drives don't gain anything from SATA III, they can be run on SATA II without issue. Put your SSD on one of the Intel 6Gbps ports and the platter drives on two of the Intel 3Gbps ports.
The ASMedia controller should not be used for anything at all, except maybe eSATA. It doesn't have an EFI compatible firmware ROM which causes all sorts of issues. You cannot EFI boot off of any of drives attached to the ASMedia controller and you cannot boot off the Intel controller if you have something plugged into both the eSATA and SATA ASMedia ports.