When you say - No Signal. What you mean. Absolutely no lights, no fans spinning, no beeps, no Bios display, no Windows?
I don't want to give you false hope, or a false diagnose. The Q6600 was highly compatible so the fact you're saying it's not showing any life might be down to the chip itself. You've re-installed your old one successfully so we presume you know what your doing.
The first place to go and check is your motherboard manual and particularly the Supported CPU list, and the BIOS revision needed. It should be available on-line if you don't have the physical manual anymore. Assuming the board and the Bios are compatible with the chip then clearing the CMOS might bring it up. Particularly if you've moved any Bios setting from AUTO with your E2620.
Good luck