Hi Mosspigs,
This has been covered before so I'll probably be brief in my reply.
To fly as Pilot in Command on N registered aeroplanes in the USA you will require an FAA PPL at the very least. These come in two flavours.
One that is based upon your JAA PPL (assuming that is what you have) and the other that is a full FAA PPL. See the following link for further information.
Licence Conversion
Obtaining the full FAA PPL isn't too stressful. Just an FAA Private Written Test and an FAA Private Checkride. The Checkride is probably the more stressful as it includes an Oral portion with the examiner which lasts around an hour but can go on for some time. The manouvers are the same except the FAA require you to demonstrate ground reference manouvers such as S Turns on your checkride. You learn these within a couple of hours.
As for a visa. If you're going to obtain the FAA PPL on the basis of your JAA PPL then you will NOT require a visa. You will require a BFR (to validate your FAA PPL) but that is not seen as training.
However if you are going to obtain a full FAA PPL, ie take the written and checkride then you will require an M1 visa as this is counted as training even though you already hold a JAA PPL (or any other ICAO PPL).
For visa requirements see the following post.
Visa Requirements for Training in USA
Hope this helps,
Charlie Zulu.