Type limitations for FAA PPL license based on JAA license?

[Runway101]

I am trying to find out if there are any limitations or restrictions for aircraft types when one applies for the FAA PPL verification that is based on a JAA license.

In other words, am I allowed to fly N-reg MD500 in the USA if my JAA license was done on a B206 JetRanger? (excuse the helicopter jargon).

Or for you fixed wing guys: Am I allowed to fly N-registered Cessna 172 in the USA if my FAA license was done on a Mooney or DA20?

Or do I need to get the type rating for everything I want to fly in the USA? Based on the FAA regulations I am allowed to fly everything up to 12,500 max takeoff weight with a pure FAA PPL, but how about a verified "based-on" FAA PPL?

Any ideas?