What is the best choice for the third period. I was thinking of the Renaissance. Is this the best choice? If so, would this be considered a period for all of Europe or was it only great in Italy. If I should include all of Europe, which countries should I specifically make reference to.
I think this is going to be answered differently by everyone that answers it. As someone else asked how do you define greatest. If you are looking at the United States one of the greatest periods in history could be considered the "Roaring 20"s", or it could be the post-war time of the 40's and 50's.
This really depends on what you are looking for in a "greatest period." Materially speaking, the greatest period in human history is today -- this is truly the golden age, at least for most people in the developed world and even in the second tier of countries, economically speaking.
If you compare the lives of average people today to those of average people during the Renaissance, for example, life is so much better today. We live longer, healthier lives. This, and our prosperity, allows us to watch our children grow up slowly, enjoying their time as children. It allows us to continue to have our parents with us, even as we approach middle age ourselves.
As for all of the art and such from those other periods, that was all for the elites. The regular person today has so much more access to art and culture than the regular person back then. This is the golden age...
You cold go with the 1960's; Civil Rights, times of upheaval are always good for historic moments. Or there is the late 18th century: French Revolution and American Revolution. Or the mid 19th: Flourish of literature in America and revolutionary icons like Karl Marx. Or, there is the Renaissance/Age of Discovery which in turn led to the era of European colonization, so it was great for technologically, artistic, and scholarly advances but bad with oppression of nations in Africa and South America, as well as North America eventually.
But, if I were you, I would use the third choice for something not Eurocentric. When you think of the "greatest periods" in history, Greece and Rome come to mind, but this is a Western centered discourse. Check out the Golden Age of Islam, which ironically occurred during Europe's "Dark Ages." The Golden Age of Islam is not just the spreading of the empire, but a flourishing of art, science and all forms of scholarship which had as much influence as math and philosophy from ancient Egypt and Greece. This example clearly shows how relative the concept of "great is;" relative to your geographical and cultural background.
Or you could go with the current time: globalization, the rise of China, ridiculous technology (ridiculous meaning exponentially more incredible than at any other time during history), the power of communication via the Internet, etc.
I don't think there is a right answer here. There shouldn't be. There is the sense that the answer to this question will depend on where you are from. If you are from Europe/North America, the tendency may be to pick three periods from those areas; particularly Europe. So, the three greatest periods is going to be subjective. But you might try thinking outside this box; easier said than done, but doable. I would doubt that, from a Native American's perspective, that the Age of Exploration/Renaissance was a "great" period.