How does the assembly from ancient Greece relate to our government today?

The only idea from ancient Greek politics that I can see as being similar would be the concept of autonomia. The Greek's idea of nomia is basically the concept of redistribution of goods. During the 800s or so, prior to the war with Persia, the Greeks developed this system of Hoplite warfare. These were the soldier that protected the Polis, or the community. They gained wealth and prestige, but they did not gain any more political power and this drove them to fight for change. They created the system of autonomia. Autonomia is the sharing of political power. They all got a say in the government and how things were run. During trials, everyone was able to attend and have their say. This is the only basic idea that is similar. The struggle with this question is that ancient Greece was not unified. Each Polis had their own government and created alliances with each other but were not ruled under one person such as a president. Sparta still had kings even after the creation of the Hoplite Oligarchy. Athens was the closest to the true democracy, their assembly was set up to where any man could come and speak their mind and vote on different issues. Similar to how we vote on different issues, so did the Athenians. So simplification, they would vote like we do.