No, once you have the conversion on the back of your CAA/JAA licence which is just an administrative excercise, all you need is a check out from an FAA instructor which will be incorporated into your checkride anyway and which is around 1hr groundschool and 1hrs flying then you are good to fly for 2 years in the USA - thats called a BFR - Bienniel Flight Review although if you rent a different airplane from a diffent place then you will need a checkride before they let you loose. It's something the insurers insist on I think. You also need to make sure that your UK licence is up to date including the UK BFR or whatever it's called here. In effect, you will have to do 2 BRF's every 2 years, one in the UK and one in the USA - It's all pretty simple really, and all thats asked is that you are sufficiently competent that you don't go and kill yourself and more importantly bend their airplane
By the way,there is no such thing as a US radio exam, it's just a quirk of the UK.
By by the way, I don't deserve any credit for the post, I just found the link that Charlie Zulu refered to, that's all - credit where credit's due but in this case it's not due. Thanks anyway!