What AMD processor supports a RAM of DDR3-1600MHz

Welcome to the forums.
There is none.
This doesn't mean you cannot use the memory though?!? I don't think you understand how memory speed works. 1600mhz rated ram is simply the highest speed it is rated to run at, it will run at 667, 800, 1066, 1333 just as well.
You can always overclock using the memory divider to get to 1600mhz after you get the rig up and running if you want to.

^ its not the processor, its the motherboard! Look up on the motherboards specfication and it will tell you if your memory bus' speed is supported, usually if it is not supported it will still work but it will downclock to the supported speed.

Sidenote, you can overclock the memory back upto its rated spec (and beyond) via the bios.

Get any Phenom II or Athlon II Processor ( Example: AMD Phenom II X3 720BE) and a true AM3 motherboard ( Example: MSI 785GM-E51) and you are good to go. They will definitely support your memory.
Why don't you tell us what components you have lying around and how much you are willing to spend so we can tell you what else you need to make it work.

Processor is just as important when the memory controller is located in the processor.