How do I connect a SAS3 drive to a SAS2-capable motherboard that only has SATA-style ports?

The 22-pin SFF-8482 adapter shown in the picture does work in this situation. As it turns out, my SAS hard drives have a feature known as "SAS Power Disable" according to Seagate's detailed specifications document for this drive.

According to the Wikipedia article on SATA:

The new Power Disable feature (similar to the SAS Power Disable feature) utilises Pin 3 of the SATA power connector. Some legacy power supplies that provide 3.3V power on Pin 3 would force drives with Power Disable feature to get stuck in a hard reset condition preventing them from spinning up. The problem can usually be eliminated by using a simple “Molex to SATA” power adaptor to supply power to these drives.

The fault in this case was that my Seasonic SSR-850TD power supply comes with 15-pin SATA power cables that apparently supply voltage to this pin, effectively preventing the SAS drives from spinning up. I was able to take the advice of this comment and use a Molex-to-SATA adapter to correct that problem. As a result, all of my SAS drives now spin up and are able to work with the LSI 2308 controller at 6Gbps speeds.

A document by Western Digital details this feature and potential issue.