The basic answer is that, in the words of Douglas Adams, it was widely regarded as a bad move.

The Athenian (and wider Greek) idea of democracy gave a vote to every adult male citizen in a direct democratic system in which the popular assembly was the main legislative power. Even in Classical Greece, many among the leisured elite regarded this as little more than mob rule. The uneducated "great unwashed" outnumbered and drowned out the educated few, and the opinion of the people was all too easily manipulated by skilled speakers with good connections and lots of money to throw around. While Athenian democracy appears to have been stable and popular, its policies were at times inconsistent, obsessive and vindictive, and the rich men who wrote history and political philosophy tended to present it as a corrupt and imperfect system.

Partly out of self-interest, the elite of Classical Greece usually preferred a system of oligarchy, in which a select few held all the power and responsibility of government. They preferred to restrict citizen rights to those above a certain property level, arguing that the rich had both the means and the education to benefit their state, while the poor had neither. Oligarchies were also considered more stable, since the wealthy were more invested in preserving the status quo, and (unlike the poor) had little to gain from war and overseas expansion.

In 322 BC, the Athenian democracy was forcibly dissolved after Athens had tried one too many times to resist Macedonian supremacy. The city-state continued to exist throughout antiquity as an oligarchy, in which a few rich familes held all prominent offices and vied with each other for the title of public benefactor through the financing of games, festivals, ships and construction works. This seems to have been the ultimate trajectory of all Greek democracies; their status as democracy was a blip on a timeline that was otherwise dominated both before and after popular rule by governments of the rich.

Later states seem to have taken their constitutional cues more from the Roman Republic than from the Athenian democracy, given that it appeared to provide a more stable and successful example of a state run partially through the mandate of the people. Modern democracies are almost without exception representative rather than direct, and are properly called republics rather than democracies.

Even in modern times, the Athenian experiment was often regarded as a failure, and other Greek political systems (such as the Spartan oligarchy) were considered a better example of how states should be run. It is only in the last century or so that Western liberal democracies have begun to see Athens as the great ancestor of their political system.

I can't tell you precisely why it doesn't recur throughout history, but I can tell you why the Greeks (specifically, the Athenians) didn't popularise it:

The short answer is that the ancient democracies, such as Athens and Republican Rome got conquered, typically by monarchies. Therefore, it wasn't really seen as a credible method of government.

The long answer is that a centralised democratic system which relied on meetings of the Athenian citizen body was not really feasible in ancient times for a number of reasons. The main one was communication - it was simply not possible to communicate quickly and effectively over long distances, and this not only was an issue for the people waiting, but placed time pressure on the Assembly's decisions. (Conversely, the Persian messenger system is praised in Herodotus - the 'neither rain nor sleet nor dead of night' passage adopted by USPS is from there)

Combined with Athens' imperial aspirations and conquests, this was a disastrous combination - for instance, the Sicilian Expedition, where the Assembly repeatedly voted troops for the doomed assault and lost a significant portion of the army - although we need to be careful on this given our main source for the period is Thucydides and he blames democracy, or at least the Assembly, for the downfall of Athens in the late fifth century.

At the risk of making a false equivalence, there were similar issues in the late Roman Republic, and it's one of the reasons why something we see in the transition from republican to imperial government were significant changes to the way in which provinces were governed, as well as the implementation of better messenger services between Rome and the provinces, plus the removal or streamlining of senatorial debate.

Despite the downfall of Athens in the late fifth century, the institution of democracy continued, but it was strained (and much less popular), particularly following the rise of Macedon, and, despite having initially retained political autonomy after Rome's conquest ultimately it would end up being subsumed into the Roman empire. Which proves the point that democracy, at least in ancient times, only worked when you were on top; conquerors, whether democratic or monarchical, are hardly inclined to question their behaviour.

The main primary sources for a period like this would be Thucydides' History of the Pelepponesian War, which takes you up to about 413 BC; if you want more detail about how the democracy functioned you should read Aristotle's Athenian Constitution (with PJ Rhodes' commentary).

The most prominent secondary reading on the subject of the Athenian democracy would come from PJ Rhodes and Josiah Ober. Not sure on who best to recommend for the later history of Athens (and Greece), unfortunately.