Don't know about local geysers, but there are two truly hot springs in the eastern US, of which both towns are called Hot Springs. I've ridden through both towns, because they are located in really nice riding areas, but have never actually stopped in either.
The first is Hot Springs in Virginia which is pretty close and surrounded by excellent riding. The resort there is called Homestead Resort.
There is also Hot Springs in North Carolina. It is surrounded by some of the best riding roads too including the 209 and the 212 which make up a stretch called 'The Rattler'. The resort there is called Hot Springs Resort and Spa.
I've even given you a route to connect the two together, while getting to ride on VA16 - 'The Back of The Dragon' and US421 - 'The Snake'.
https://goo.gl/maps/gXhgHmVEQwS2