In Greek tragedies, the action was in the movement of characters through various classes in society. But in modern tragedies, the action is more in the thoughts and emotions of the characters. It's much more mentally based. Like the writer of the fifth tragedy reading said, summaries of Greek tragedies sound a lot more exciting than summaries of Shakespeare's tragedies. But Shakespeare's work is powerful because you see the struggles not between nations or power holders, but between a man and himself, or between a man and society at large.
As Arthur Miller said, the tragedy of the common man is much more powerful today than the tragedy of people in high places. Besides, the "flawed" heroes of Greek-style tragedies are either noble, flawless people who are victims of ridiculous situations (Oedipus) or just dirt bags to begin with who seemed like cool people because they were lying (Enron and friends). Bill Clinton, it's true, seemed to fit the description of a great man who made a mistake, but his story doesn't qualify as a tragedy because, well, look at him. It didn't exactly end in tragedy. He remained president and he's now a big political guy who Democrats love.
So what gives us that tragic feeling nowadays? When the action is in the characters' emotions, their personal crises, their brains. It's more inwardly focused. No longer do we weep when a great leader falls. The kings are dead. We are coming into the age of the people.