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Burgess Falls (in White and Putnam counties not Cumberland) is very nice, but I don't know that I would go out of my way with the closures. If you're going through the area already you might want to try Cummins Falls instead. It's been listed as one of the best swimming areas in the country so it's a great place to go, but does get a ton of people every day.
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It's a bitch of a hike as well. If you don't like long hikes, down a huge hill, cross a river three times, and then more walking before you swim, stay away. Now, if that didn't scare you off, it's one of the most beautiful spots in the state, and worth the walk if you're in good shape.
Another option that's a little closer is Ozone Falls. It's where the final scene from The Jungle Book was filmed, and is an easier walk.