Hello,
The T610 supports 12 DIMMS up to 192GB of RAM with Registered ECC memory. It also supports a max memory of 24GB with Unbuffered ECC memory.
You can refer the following link for the memory. www.memorystock.com/.../DellPowerEdgeT610DDR31333MHzRegECC.html
In order to upgrade to the max (192GB) you can go with 12 sticks of 16GB PC3-8500 1066MHz Registered ECC (Quad rank) memory.
Detailed information of the memory including the rules of installation can be found in the following link. (page 30 - 32)
http://www.dell.com/downloads/global/products/pedge/en/server-poweredge-t610-tech-guidebook.pdf
Note: Registered ECC and Unbuffered ECC cannot be mixed.
If the existing memory which you have installed is ECC, then you can replace them with Reg ECC memory in order to upgrade to max or to upgrade more than 24GB of memory.
To check what is installed in you server from a remote location you can refer the 2nd post.
Good luck with the Upgrade!!
BR,
John