Based on my limited knowledge of history, I can probably say that the civilisations one such as me thinks have had a grand impact are more likely to be in Civilization V- the game is not aimed for historians.
People arguing that so-and-so nation should be put in because the game will sell more is silly, how many of you bought the game so you could play as your nation? You just go for the nation in accordance with how you want to play. If game-sales were based on nationalism and patriotism alone the game would simply be full of modern nations (I was tempted to make a Serbia joke here, but decided against it. No offence intended to my Balkan brothers

).
American Civilisations
United States of America - confirmed
Incans - The first grand nation of the new world the Spaniards came across (to my knowledge), heavily romanticised and the atrocities done against them is common knowledge.
Azteks - According to the earlier posted graph, they've been every civilization game, perhaps they are well known in the USA.
Mayans - This civilisation has had a lot of publicity recently with the end of their calendar, various films, perfumes etc. for some reason this civilisation has had a lot of publicity of late.
European Civilisations
Germany - confirmed
France - A great deal of people know of this country and of its power, present and past.
Great Britain (or the UK) - This is the same case as with the French.
Spain - " "
Russia - " "
Roman Empire - Immensely famous all over the world due to their relationship with Christianity, the creation of Latin, conquering and assimilating most of Europe and other things.
Ancient Greece - A common source of tales and legends which many people grew up with, most people will know of at least one Grecian philosopher or invention.
Asia
China - A great deal of people know of this country and of its power, present and past.
India - " "
Ancient Persia - This region is currently very well known due to Iran's involvement in wars and their fairly recent toppling of the monarchy, their art styles are still evident in your average household, most people will own or know of Persian rugs.
Mongol Empire - Genghis Kahn is still told of and is heavily romanticised, the remnants of his empire caused the creation of many other nations such nations will obviously be more familiar with him.
Japan - Well known in America due to their involvement in the War and throughout the world thanks to their involvement in modern culture and samurai.
African Civilisations
Ancient Egypt - for similar reasons as Ancient Greece
Zulu - a well-known and again romanticised nation due to their aggressiveness and involvement in the Zulu war
Obviously, there could be many other countries included, but perhaps none are as famous as the above (this depends on where one lives naturally, I am speaking from an English perspective). Unfortunately, I am out of slots and I fear Europe would be over represented with the Ottoman Empire involved.
I've no doubt civilisations such as Portugal, Mali and Babylonia will be included in expansion packs.
Sorry if my explanations are difficult to read or ill-explained, I am very tired.