There's nothing official about them that screams "Greek" in particular, though you could play them that way if you wanted. They're just a generic ancient fantasy empire, more of a placeholder for whatever evil ruins the campaign demands than anything specific. I've also heard it suggested that they could be Egyptian-like or Mesopotamian-like in culture, and James Mishler suggested they could be the Taymorans
. The problem with making them Greek is that the real Greeks never really had an empire, except for Alexander's short-lived realm, so if you combine Greek architecture and culture with anything like a long-lasting empire you end up with something more like Rome or the Byzantines instead. Anyway, there's nothing in the maps or illustrations of Turathi ruins that I've seen that suggests Greek, Mesopotamian, Egyptian, Roman, or any other recognizable historical style. They're a high fantasy civilization with no overriding theme other than "diabolic."
The most complete general source on Bael Turath is Player's Handbook Races: Tieflings
by Matthew Sernett, which still isn't a lot of detail, but it covers a few named cities and royal houses of the ancient empire, and the basic story.
Basically, Bael Turath started out as a fairly normal empire, not particularly good nor bad, but after growing much too large and lasting rather too long (centuries at least), they started to feel the pressures of being overextended and decadent. There were civil wars, coup attempts, secession attempts, famine, plagues, and so on, the worst of the empire's nobles (including the emperor) prayed for the secret of ensuring their empire would be lasting. They began having dark dreams, sent by the archdevils of the Nine Hells. Soon they decided to conduct an evil ritual, the Bloodfire Moon (a very typical 4th edition name, following the same compound word formula that brought us the Feywild, Shadowfell, the fire archon blazesteel, the vampire spawn fleshripper, and the goblin blackblade), that would tie their bloodlines to the Hells forever. Those noble houses that refused to take part in this ritual died in battle or were sacrificed on the altars of the Lords of the Nine. Each noble house pledged itself to a different devil with pacts with flowery names like the Scarlet Claw of Hunger, the Heart of Fire and Iron, the Cage of the Unnamed Master, and the Iron Crown of Madness. Those who made the pacts became tieflings. With their hell-bought powers and newly found racial solidarity, Bael Turath flourished for centuries to come, growing to encompass half the world and even extending into the Underdark and other planes of existence.
Even very late in the "tiefling era" of Bael Turath, most of its citizens were normal humans or other nonplanetouched races; only the nobles were tieflings. The tieflings today (in the generic 4e world) are all descendants of one or more of these ancient noble houses, though they may not know which ones.
Eventually Bael Turath came into conflict with another great empire, the draconic (not just dragonborn, but also full dragons) empire of Arkhosia. They skirmished, they warred, and finally they wiped each other out in the War of Ruin (some say due to an out of control apocalypse spell, as described in Monster Manual 3
). The ruins of both civilizations can be found throughout the world, since many territories changed hands multiple times, being ruled by Arkhosia or Bael Turath as the respective empires waxed and waned. Again, it's mostly a matter of convenience, so the DM can put either kind of ruin anywhere in their campaign world without having to worry about history or geography. A given dungeon could have a Turathi level, an Arkhosian level beneath that, another Turathi level beneath that, and another Arkhosian level beneath that if you really wanted (and a level dating back to the more recent Kingdom of Nurath on top).
Named cities of Bael Turath include Vor Kragal (from Dragon #364), Vor Rukoth (from the setting book of that name), and Harrak Unarth (from Dragon #368). It's implied that Vor Kragal was the empire's capital at one point. Vor Kragal was buried under ash because of the magic of an elementally-aligned noble house (Zolfura - or possibly a dragon assault had something to do with it). Harrak Unarth was pulled into the Shadowfell's Domains of Dread (essentially Ravenloft) by the doomed love and betrayal of its rulers. In Vor Rukoth a portal to the Nine Hells went terribly wrong and the city remains overrun by devils to this day.
Monster Manual 3
claims that skulks were once a slave caste in Bael Turath gifted with supernatural stealth by the demon prince Graz'zt as a reward for rebelling against the devil-aligned nobles. Skulks have been the sworn enemies of tieflings ever since.