Vijayanagara is better known as Hampi, and it is amazing, the picture doesn't do it justice. The whole city was carved out of enormous boulders, which make up most of the surrounding landscape. It is incredible to behold the scale of it. Highlights include a camera obscura, elephant stables, a water-cooled palace, a tuned 'musical temple' and miles upon miles of grand, ruined streets.
Angkor (now central Cambodia)
http://en.wikipedia.org/wiki/Angkor
Hugely significant for the language and culture of mainland Southeast Asia.
Dali, Nanzhao (now Yunnan, China)
https://en.wikipedia.org/wiki/Nanzhao
Important as a highly developed kingdom with influence over a large area that acted as an early crossroads between China, India, Tibet, and non-state peoples of Zomia.
Various cities, Shu kingdom (now Sichuan, China)
http://en.wikipedia.org/wiki/Shu_%28state%29
Rediscovered in the 1980s, purely only known by literary reference in ancient Chinese texts, this civilization is known to have had highly developed bronze working far beyond the capacity of any other known in human history. We really know little about them as they did not have a written language, other than they had some form of ritual or belief system involving masks, large and ornate metallic trees and bird figures.
Various cities, Champa (now central and southern Vietnam)
http://en.wikipedia.org/wiki/Champa
Champa was the pre-Chinese Indian kingdom of Vietnam. It was finally conquered in the 15th century, though its descendents remain. Huge Hindu temple complexes remain which are spectacular.
Jambi, Srivijaya (now eastern Sumatra, Indonesia)
https://en.wikipedia.org/wiki/Melayu_Kingdom
Interesting because it's huge and mostly unexcavated - potentially a second Angkor.
... and probably at least 3 or 4 early (eg. Mon) sites in Burma.
That's a very good question, and I'm also quite curious. A quick look at the Wikipedia article for Skara Brae mentions that pottery, flint and ashes have been found at the site which certainly suggests they had fire.
I'm pretty sure I read somewhere that they used dried seaweed and dried animal dung. Maybe peat as well - it's still a fairly common fuel in the wilder parts of Scotland.
Skara Brae is in an amazing part of the world for neolithic monuments - Maeshowe, the Ring of Brodgar and lots of others.
Edit: The Viking graffiti in Maeshowe is pretty special too...
I'm not sure what they could have used that would leave behind ashes but it is possible to use animal fat as a fuel source. If you could get enough there to be useful I'm not sure though.
Skara Brae is amazing: it's a very freaky feeling standing in something still recognisable as a home after four thousand years.
But it is in no way a city. The article admits as much, calling it 'one of the best preserved Stone Age villages in Europe'. Even the word 'village' is generous; it's a hamlet.
I guess '34 Lost Settlements' wouldn't have the same snap as a title.
https://en.wikipedia.org/wiki/Vijayanagara