I'm sorry, but you're incorrect. The American government is a hybrid-system of both democracy and a Republic. We aren't Rome.
You could argue that we "used" to be a Republic, in that anyone could vote...as long as they were white, male landowners.

America is a Democracy in that all it's citizens are allowed to elect and have a say in the nomination of it's Republic representatives. It's not the same as being Republic citizens and voting, though there are still elements of that; such as in electing Supreme Court Justices.

People like Neil Bortz like to rant and rave (as much as I love his show) about how we aren't a Democracy and how the founding fathers dreaded it. This line of thought isn't so much selective history, as much as it is nostalgically recalling a history that never existed.