The most notable characteristic the United States has adopted from Ancient Greece is the idea of a democracy. Democracy, or the Ancient Greek demokratia, literally means "people's power," and refers to a system of government where the people are in charge of the decisions. There is some ambiguity to the actual meaning of democracy- who really has the power? Who is part of the "people?" Everyone, or only the wealthy elite? It is difficult to say who actually had voting or decision making power in Ancient Greece as there is no regular census data, and most of the contemporary first-person accounts of Ancient Greek politics are from the perspective of a wealthy, educated, elite person.
In the United States we experience some degree of ambiguity in our practice of democracy, too. While every American citizen has the right to vote, many do not, and their voices go unheard in the decision-making process. There are also criticisms of the democratic republic system, whereby populations elect a representative to vote or act on their behalf in the government. People may be unhappy with their chosen representative if their ideals differ from the elected official's, and there are criticisms of how easily some politicians may be swain by bribery.
Another aspect of culture the United States has acquired from Ancient Greece is architectural style. Many government or public buildings and monuments are built in a style called Neoclassical architecture, meaning it incorporates new symbolism and motifs but is heavily based on Classical (Ancient Greek) architecture. If you look at Ancient Greek structures, you will likely see lots of columns, domes, and decorative carvings. These, too, are evident in the architectural styles popular in the United States during the 17th, 18th, and 19th centuries.
The fact that Neoclassical architecture is employed in the style of many public buildings like courthouses, banks, and government offices serves as a visual reminder to where much of the United States' sociopolitical ideology comes from.