Depends, some games like fewer threads with a high IPC, and others like multiple threads, but most games don't use more than 4 threads these days. But yes you will see a performance degrade of you get lets say a Xeon of the same architecture but its lower clocked but has 8 cores and 16 threads. I personally would just stay with your i7 930 and its current overclock, 4 cores and 8 threads and triple channel ram is still pretty goods these days.
A 4790k at stock 4ghz 4 cores 8 threads will be faster in most games than i7 5820k at stock 3.3ghz 6 cores 12 threads, they are the same architecture so a lower clock normally means lower performance, Overclock the i7 5820k to 4ghz and it will perform identical in lightly threaded games, though if you want to stream and game at the same time, the 5820k even at its 3.3ghz can still edge out the 4790k at 4ghz. The 5820k 12 threads can help with multi tasking.
Now if we compare a brand spanking new CPU such as a i7 6900k 8 core 16 thread chip at 3.2ghz vs your i7 930 at 3.8ghz, the 6900k will probably beat your 930 in lightly threaded apps to since is newer architecture and has faster IPC.
The high thread count CPU's that will most likely work in your motherboard are probably not worth touching for gaming, even if its only 1 or 2 generations newer, it will loose to you'er 930 in gaming since most games don't benefit from more than 8 threads.
Honestly games have not pushed much into using multiple threads, only game I can think of is Battlefield One, anything else gaming wise is hardly touches my CPU.