There's a pay lake at the entrance to Meeman-Shelby Forest called Lyle's Lake that has huge grass carp in it.
The old lady that used to run the place died, and her son or daughter (forgot which) run the place now. They don't have as much free time as she did, so it's not open a whole lot.
*I THINK* I remember the lake being open Memorial Day through Labor Day, and it was either weekends only, or Fri/Sat/Sun.
Look up "Shelby Forest General Store" on an online map. Head north from there towards the entrance to the park, and it's on your right about 200 yards down with a big white sign in front of the house.
I've also been told that Casper Lake (the big lake in Edmund-Orgill golf course in Millington) has carp in it, but I've never looked for them. It's relatively big and has a public boat ramp that I think is free.
Those are the only two that jump to mind, but if I think of others I'll let you know.