I think you might need to explain your question in a bit more detail. First, when you say "Ancient Persia", do you mean classical antiquity, as in the Parthians, or earlier (Seleucids) or later (Sassanids)? And what do you mean by "great city"? Because there were a number of quite important cities in each of the time periods. Persepolis, Babylon, Seleucia, and Ctesiphon were all large cities in their time. None of these probably reached the sheer population of Rome in their respective heydays, but that's not really a fair comparison, because neither Athens nor Carthago got close, either.

If you are referring to the Achaemenid Persian empire, the main reason is that this is a vastly different polity from the other three that you mentioned. Athens, Carthage and Rome were all city-states, which grew around a single urban centre and came to dominate other regions either through the establishment of overseas settlements or through the absorption or subjection of other polities. The Persian empire, on the other hand, grew out of the territory of a tribal group without major urban centres of its own (though heavily influenced by the more urbanised culture of its Elamite neighbours). It took over various regions that contained major cities (such as Mesopotamia, Phoenicia, Egypt and Anatolia) but did not itself originate in a single large settlement like Athens or Rome.

Since imperial power in Persia essentially revolved around the person of the king and his family and retinue, it was important for the king to be seen across his empire. Apart from military campaigns that brought the peoples of the empire to the king, and the king to the edges of his empire, this sort of visibility was achieved by creating a travelling court. The heart of Persian rule moved between several major centres in the Achaemenid heartland - Sousa, Ekbatana, Pasargadai and Persepolis. The latter two were purpose-built palace settlements, without much urban life beyond the services supporting the imperial palace.

This is not to say that the empire did not sustain city life. The great ancient city of Babylon flourished under Achaemenid rule, remaining the major centre of Mesopotamia throughout the period. The cities of Phoenicia also seem to have fared well. Cities along the network of Royal Roads received a boost to their economy from traders and travellers passing through. To the Greeks, the epicentre of Persian power was always Sousa, despite their awareness of the other settlements in which the king sometimes resided; for their immediate political purposes, the hub of Persian authority lay in the Lydian city of Sardis in Western Anatolia. However, given that much of the Achaemenid government structures were simply superimposed upon existing ones, and that the reach of Persian power was mostly felt through the presence of garrisons in captured towns across the empire, there was no pressing need for the Achaemenids to establish new cities of their own.