As a former AAU Freestyle wrestling champion, I've read a lot of assumptions about wrestling styles. Some have been fairly accurate, but none have been 100% correct.
I hope to clear up the differences between the two styles of wrestling being discussed in this thread which are Greco Roman (not Roman-Greco) and Freestyle. We will not be discussing scholastic (high school) or collegiate wrestling.
Greco Roman wrestling forbids the use of the legs for defense or offense therefore most takedowns do come from upper body throws or suplexes, but that does not make up the entire arsenal of a Greco Roman competitor.
Talented Greco Roman wrestlers use their hips with weight and momentum to take down their opponents with arm whips, whizzers, and other sneaky techniques.
Once on the mat, the bottom wrestler will usually flatten out to his stomach as the top wrestler attempts to turn the bottom man, often with a gutwrench, which is clasping both hands around the man's waist from behind, squeezing, and trying to roll over.
Other tools for placing the bottom wrestler on his back would be: arm bars and a half nelson.
Freestyle wrestling allows the use of the legs and therefore opens up the arsenal of the competitors. The goal is the same as in Greco Roman: put your opponent on his back and stay off of yours.
It would be impossible to discuss all of the options available to Freestyle wrestlers. There are too many.
If you'd like to know about anything specific, post a question in this thread, and I'll do my best to clear it up. It's cool to see an intelligent discussion on an important discipline in MMA: wrestling.