Since the first 2 posters covered political ties and architecture, I'll focus on art & philosophy. In addition to finding the fundamental basis of our republic, the seeds of our ethical codes and ideas of individual thought can be found in writings of Plato and Aristotle.  Many modern educational techniques are rooted in the Socratic method. In fact, "Socratic seminars" are a highly praised strategy in the classroom. Also, classical Greek tragedies have influenced our contemporary theatre. The Greek dramatic tradition is alive in nearly every theatre in America.

One visit to the capital, Washington, D.C., will answer this question.  For, there much of the architecture of the buildings that house our lawmakers that evidence the influence of Greek architecture with their Corinthian columns.  The Treasury Building is also an example.  The Greek Revival architecture dominated America from 1818-1850.  Many of the plantation homes in the South are lovely examples as well as some stately buildings in the North such as the Towne & Davis & Frazee building, the Custom House, of New York City and the Custom House of Boston.

Many important principles of American government can be traced as far back as 2500 years ago, originating in the civilizations of the ancient Mediterranean civilizations of Greece and Rome. To answer your question I will respond to the Greek influence. The word democracy is derived from the Greek word demos meaning the people and kratos meaning rule. Translated  is defined as a system of government where by political power rests with the people. Having said that it must be understood that this political philosophy had nothing to do with equality.

The idea of democracy began in Athens as an 'oligarchy' where a few people held political power for the majority. In ancient Greece the oligarchy was made up of a council of the wealthiest male citizens. Women were not considered citizens, therefore were not permitted to participate in the political process. However, the demos-kratos that developed in Athens did plant the seeds of the democracy we understand today in the United States.