Final answer:
The Greeks influenced American democracy primarily by pioneering citizen participation in governance. The system of direct democracy in Athens allowed citizens to vote on government matters, directly shaping the democratic principles that the United States would later adopt.
Step-by-step explanation:
The most important way in which the Greeks influenced American democracy was by allowing citizens to participate in government. The ancient Greek city-state of Athens is renowned for developing a system of direct democracy where citizens could vote directly on laws and governmental matters. Although not all residents of Athens could vote, this represented a groundbreaking move towards popular participation in governance, which has deeply influenced the American political system.
The United States is considered a republic, which means it's a form of representative democracy. The system in the U.S. was influenced by both Greek democracy and the Roman Republic. The founders of the U.S. studied these ancient systems and integrated the concept of citizens electing representatives to act on their behalf into the fabric of American governance. The Athenian model and the notion of popular sovereignty have left a lasting legacy on how democracy is practiced in the U.S.