You may be struggling to rent a big Cessna or a Bonanza (just as you would in the UK and I assume Oz). You'll have much more chance with a 172, PA-28, DA-40. Where are you planning to go in the USA ? and what do you want to do ? I've rented all over and it isn't that hard. I assume you have the appropriate license.
Maybe give better advice if you narrow the question down.