The geographic region of North America that best matches this description is the Great Plains.

Here's why:

* Rolling flatlands: The Great Plains are known for their vast, relatively flat landscape.

* Many rivers: The region is crisscrossed by numerous rivers, including the Missouri, Mississippi, and Arkansas Rivers.

* Broad river valleys: The rivers have carved out wide, fertile valleys that are often used for agriculture.

* Grassy hills: While the Great Plains are predominantly flat, there are also rolling hills and buttes, which are often covered in grasses.

Let me know if you'd like more information about the Great Plains!