The Intel H61, H67, P67 and Z68 chipsets support both Sandy Bridge (2nd generation) and Ivy Bridge (3rd generation) processors; contrary to Wolfshadw's statement. The support for 3rd generation processors for these chipsets were provided by the manufacturer via a BIOS update. Without the BIOS update, 3rd generation processors are not originally supported.
The BIOS update must be designed specifically to support the newer CPU's, otherwise a BIOS update for another purpose will still not support 3rd generation.
Whether Dell has pushed an update supporting Ivy Bridge I'm unsure of, and cannot guarantee compatibility. However the
latest BIOS update for the Dell Optiplex 390 from Dell, which was released on 10/30/2013 and last updated 3/21/2014. The BIOS update changelog/"Fixes & Enhancements" states "Updated CPU Microcode" which indicates that new CPU's are supported; whether they are referring to Ivy Bridge is debatable, and as mentioned earlier I cannot guarantee compatibility (personally I believe they are referring to Ivy Bridge.)
However I noticed a web site via Google called "TechBargains" selling the
Dell Optiplex 390 with an Ivy Bridge (3rd generation) CPU. If this is the case, they most likely have the latest BIOS, and it would confirm that the latest Dell BIOS update supports Ivy Bridge. However I'm not too sure on how creditable the TechBargains source/website is.
Again I cannot guarantee compatibility, but you could update your BIOS to the latest and go down to your local computer shop, and if they have a spare Ivy Bridge CPU, perhaps you could test it in and see if your system boots. If it does, then that confirms the BIOS update was designed for Ivy Bridge.
All the best.