You can't fly directly on your JAR licence.
You have to get a US licence, there are two ways to do this
1 Get a "piggyback " licence which is a US licence issued on the back of your JAR one. As long as your JAR one is valid then your US one will be. The link above details how to do this.
2 Take a US written and flight exam and get a standalone US licence. This will also require a US medical. The advantage of this is that you can add US ratings such as instrument etc on to it whereas you can't with option 1.