The Athenians would not have recognised today's democracies as democracies. We lack the institution that stood at the centre of the Athenian system - the Assembly of all adult male citizens. Our democracies are built around a parliament of elected representatives instead. The only time our modern system comes close to the way things were done at Athens is during a general referendum, when people get to vote directly on issues proposed by the executive branch.

Athenian democrats would have strongly disapproved of our modern democratic systems of government for a number of reasons. The main one is the fact that, in many countries, we elect our representatives both for the legislative branch (parliament/congress) and the executive branch (president). The Greeks thought elections were very undemocratic. Those with money, connections and rhetorical skills would always be able to sway the outcome in their favour. For this reason, officials at Athens were selected by lot from all citizens above a certain property level. This was considered the most fair and democratic way to choose. To counter the possibility that those randomly assigned to important posts were staggeringly incompetent, most aspects of government were managed by a board of several officials whose duties were shared. The only exception to the rule of selection by lot was the board of ten generals, who were elected directly by the Assembly.

Another reason would be the fact that we elect our representatives for a period of four or five years. The Athenians would have despised a system that gave its leaders so much opportunity to entrench themselves in their positions of power. At Athens, officials served for one year only. Many offices could not be held more than once. A place on the Council (the executive body of citizens that handled everyday affairs and led Assembly meetings) lasted for one year and could only be held twice in a citizen's lifetime.

These two points create the awful - in Athenian democratic eyes - combination of officials who come to power through rigged elections, and then serve for a long time without being directly accountable to the general population. They would think we were practically asking for someone to set themselves up as a tyrant.

A final thing they might have found abhorrent is the fact that we let all adult residents vote. No Greek city-state granted any political influence to women or resident aliens. The same of course went for slaves, and they would presumably be baffled that we don't have any.

Probably the only aspect of government that Athenian democrats would have recognised and liked is the jury courts that exist in the US and UK, which are a much closer equivalent to Athenian democracy than our modern democracy.

That said, it is possible that political thinkers of a more oligarchic bent (like Plato and Aristotle) may well have approved of certain aspects of our indirect democracy, including its bias towards an established elite of wealthy career politicians.

First off, I am happy that you distinguished between Ancient Athenian and Greek, because they are not one in the same.

Athenian democracy is very interesting because it was very direct. You had private citizens, ἰδιώτης, idiotes, that would proceed to the Pnyx, the assembly gathering area, and discuss everything from foreign policy to how much someone's getting buggered, and therefore shouldn't have the right to speak in front of the assembly. Athenian democracy, in fact, was so focused on the private citizen that "democracy" is actually an insult given to describe this system, which was pitched to the Athenians as "Isonomia", the equality of law. Democracy literally means the power of the people, which has undertones of a demagogue (looking at you, Creon) who is able to sway the voters (like Donald Trump) to vote in a jingoistic and chauvinistic way. (Chauvinism is only pro-men when "male" is used as a modifier, as "Chauvinism" only means "excessive or blind patriotism" ). So "democracy" only derives from the existence of "demagogues", or leaders of hoi Polloi, the many. Athenians would immediately recognize Donald Trump as something that existed back in Athenian times. In fact, Athens lost because of the jingoism of the leaders that arrived after Pericles, who was able to balance the needs of Athens with the voting power of the people.

I think looking at Athens is quite unrepresentative. Athens was enormous. 30,000 citizens (free males who were born in Athens to Athenian parents over the age of 18). Athenian democracy was supplemented by various magistracies and year-long offices for the administration of the city. In the typical city state, the citizen body would be about 1,000. What does this mean? It means that this form of direct democracy would make much more sense. There would be about the size of the U.S. House of Representatives meeting to approve and discuss proposals made by the equivalent of a U.S. Senate committee, or propose their own, if their laws allowed them to.

I think therefore, the high enfranchisement of the citizens of the Ancient Greek world would look at puzzlement of the lack of political involvement that the average citizen of the United States.