Rar Rar Rarr!
What does Pig-American sound like? and variations thereof are very popular questions on ask.metafilter:
How does one speak "pretend English"?
and
What does American English sound like to people who don't speak english?
There seems to be some consensus around American English sounding like "rar rar rar." Anything where you pronounce a lot of hard Rs. Also saying "Howdy, y'all", "awesome," and requesting cheese, although those observations are a little off-topic.
All of this sounds right on to me, as I pronounce all my Rs as hard as humanly possible, actually do say "howdy" (although it sounds more like "haddy"), use the ubiquitous "awesome," and, like any good American, I love to put cheese on everything.
However, as others point out, any attempt to codify the sound of Pig-American is flawed because the US is a big place where lots of people don't pronounce their Rs and don't say "howdy" or "awesome." (I think the cheese thing is probably universal, though.)