Would there be any performance issues running a 3570k with a decent OC, and 1066Mhz RAM?
No there shouldn't be any real problems. Overall DDR3 RAM as a whole is faster and has more bandwith than older RAMS and speeds don't really make a HUGE impact on performance.The only time where speeds seem to make a difference is with AMD's APU's when you are using integrated grpahics.
RAM is so cheap if you can afford 1600 go for it but again I really don't think it will make a huge performance difference. I have two builds the one below in my sig with 1600 speeds RAM and another with an I5 2500k, 7870, and 4 gigs of Ram at 1066. I can't say I see any noticable difference in when gaming. I still get good FPS and great performance.
Thanks. For gaming, is faster RAM worth it at all?
Short answer: I suppose you are really asking "should I use this DDR3-1066 I already have or buy new memory?" If you have enough DDR3-1066, use your current memory, it will work fine. If you don't have enough memory, buy more of whatever makes for a good deal (probably will be DDR3-1600 speed) and put it in with your existing memory. It will downclock to DDR3-1066 and work fine.
Long answer: It depends on the chip and if you are using a discrete graphics card or not. Gaming with a discrete chip doesn't tax the memory subsystem a whole lot and you won't see much of a performance difference between DDR3-1066 and above. DDR3-800 and slower will start to cut into performance but as far as I can tell nobody sells DDR3 slower than 1066 so it's a moot point. Here is an article explaining memory scaling on an Intel i7-2600K, which is very similar to your i5-3570K. Now if you are using integrated graphics, completely different story. That is really dependent on memory bandwidth. I however would keep the DDR3-1066 and spend the money you would have spent on faster RAM to buy a discrete GPU as pretty well any remotely modern discrete GPU will be much faster than an Intel IGP.