H61 is sandy bridge chipset. It supports only 2nd gen intel cpu natively.  If there is BIOS upgrade, then the chipset can support 3rd gen intel cpu.  your issue is due to Dell not providing such BIOS upgrade, therefore you can not run 3rd gen cpu.  Dell prefers customer to buy the next generation optiplex pc that has ivy bridge chipset (H7#) which supports 3rd gen cpu.
Just recently bought a Dell Optiplex 390 i3 2120 because my old system died. I changed the cpu to a i5 3470 but got no post, the Intel H61 Express cpu list states it supports it. I updated the BIOS to the latest A14 which just made it boot cycle and now wondering if the mb just doesn't support the cpu and if there is any way to get it working?
Dell optiplex 390