In anticipation (at the time) of Hurricane Rita, the Western Mail had a feature on assorted lost cities ... here's excerpts of the ones in our purview:

[...]

Babylon
MOST famous for its (allegedly mythical) hanging gardens, the city of Babylon had actually existed for nearly 2,000 years before they are supposed to have been built under the rule of King Nebuchadnezzar II between 605 BC and 562 BC.

This process of beautification made Babylon one of the most impressive cities of the ancient world, but its decline began after it came under the rule of Alexander the Great in 331BC. After his mysterious death eight years later, his empire was divided between his generals, who were soon at war. Babylon bore the brunt of the fighting and its residents emptied the city. By 141BC, Babylon was desolate.

Most of the existing remains lie on the east bank of the Euphrates, in modern day Iraq.

[...]

Troy
FOR many years, the existence of the city of Troy was not believed to extend beyond Homeric legend.

Then in the 1870s, the German archaeologist Heinrich Schlie-mann discovered the ruins of a series of ancient cities, dating from the Bronze Age to the Roman period, while excavating a hill at Hissarlik, in north-western Turkey.

Further exploration at the site revealed there were, in fact, the remains of at least nine cities built on top of each other. A mammoth investigation by a team of archaeologists under the direction of Manfred Korfmann, starting in 1988, found a series of Bronze Age artefacts and a deep ditch along the periphery of the ruined city.

Despite Korfmann's work, Troy's authenticity is still not universally accepted among ancient historians, with many disputing the commonly held perception that the city is Troy is synonymous with the Hittite city of Wilusa.

Carthage
FOUNDED in 814BC, Carthage used its position on the western shores of the Mediterranean to become the most prominent commercial power in the region, a mantle it held for several centuries.

The city became rich through trading in silver and tin ore, but it became notorious among its neighbours for its child sacrifice rituals. Carthage's enviable position and wealth made it a target for opposing empires, and the city rulers became engaged in wars for several centuries.

Carthaginian power held out until 146BC, when Roman soldiers invaded, slaughtering many of the residents, burning the harbour and razing the city. It was rebuilt and continued to be inhabited for several centuries before eventually being overrun by Islamic forces in 698AD. It is now a popular tourist attraction in modern day Tunisia.

[...]

Atlantis
THE story of an affluent island sunk to the bottom of the sea was told by Plato around 360BC in his dialogues Timaeus and Critias, and has sparked controversy and debate among scholars ever since.

While telling the stories, Plato repeatedly stated that the story was true, encouraging speculation that it was actually a fictionalised version of a real story.

In Plato's version, the island was the domain of the sea god Poseidon, and its rulers exercised influence beyond its shores into Europe and Africa.

The climate and power of the island produced great wealth, making the residents greedy and corrupt, leading to the city's punishment by the gods. It has since been suggested that the story was inspired by the submersion of the island of Thera in the Aegean Sea by a volcanic explosion in 1500BC.

Evidence of a sophisticated culture, probably part of the Minoan Civilisation, has been found at the site of the island.

Now considering that the context of this article is 'cities that weren't rebuilt after natural disasters', it seems somewhat strange that neither Pompeii nor Helike made the list ... but I'm probably just picking nits.