For the majority of people there is no justification for upgrading the internal drive bays to SATA III, as I mentioned I got the parts secondhand and therefore cheaper and did it more because I could then really needing to. Remember that traditional hard disk drives simply cannot go fast enough to need SATA III speed, only an SSD drive will get a benefit. However here are the details on how to do it.
If you have a Mac Pro 2006 to 2008 model Mac then this design has a cable going from the drive bays to the logic board and the connection on the logic board is a standard miniSAS connector. It is therefore possible to unplug this cable from the logic board and extend it to a PCIe card with a miniSAS connector and this PCIe card would then act as a SATA III controller. See this article for more details http://blog.macsales.com/12247-upgrade-your-06-08-mac-pros-internal-bays-to-sata -3-0
I would not necessarily get the PCIe card referred to by that article as according to the MacSales website it does not support booting in to OS X. I would instead get one of the following.
https://www.startech.com/uk/Cards-Adapters/HDD-Controllers/SATA-Cards/PCI-Expres s-SATA-III-RAID-Controller-Card-Mini-SAS…
https://www.attotech.com/products/adapters/sas-sata/6gb-pcie-30/ESAS-H644-000
https://www.attotech.com/products/adapters/sas-sata-raid/6gb/ESAS-R644-000%20ESA S-R644-C00
(The last in the list above is a hardware RAID controller and therefore more expensive.)
I know from personal experience the first two work fine for both booting in to OS X and even Boot Camp although you need to first install the Windows driver to enable Boot Camp booting. This can be done by first installing Windows and the Windows driver via the original SATA II interface and then swapping it to the SATA III interface.
If you have a Mac Pro 2009, 2010, or 2012 model then the process to upgrade the drive bays is a bit more complex and expensive. It is not possible on these models to simply disconnect them from the logic board. Instead you need to buy special drive sleds which bypass the built-in SATA II connection but still utilise the built-in power connection, you then use a miniSAS fan-out cable to connect each of these special drive sleds to again a PCIe card and again the same choices as I listed above would work. For the special drive sleds and miniSAS fan-out cable you need to get these from here. http://www.maxupgrades.com/istore/index.cfm?fuseaction=product.display&product_i d=189%20
I have a secondhand Mac Pro 2010, I also got a secondhand MaxUpgrades set of drive sleds and cable, and got a StarTech PCIe card. With a SATA III SSD drive I do see an improvement in speed and I have it working in both OS X even in El Capitan, and also in Windows 10 via Boot Camp.