I tend to game for narrative immersion. One of the easiest (for me) narratives is an historic one. Here is my list of games to play through world history with 4 players.
Start out with Pax Romana. Second edition is supposed to be coming out sometime (it's on GMT's P500 list so once enough people pre-order it will be published). Pax does have some problems with the physical materials. The money is basically bunk. If you flip your coins over accidentally then you don't know how much cash you have (different denominations on each side). It does do a great job of recreating a strategic view of the period. One thing to keep in mind is the time scale. With 25 year turns, a lot of unexpected "random" things can upset your plans, but it's still great fun.
Now that the rise of the roman empire is broadly covered, it's time for the fall. I actually have 2 games in this category, but one (Barbarian, Kingdom and Empire) is currently out of print. If you're interested, Decision games does have a pre-order program to spur their reprint.
Britannia is one of the best games to cover the transition from the Ancient World to the High Middle Ages (although much of the game is set in the Early Middle Ages). It focuses on the various invasions of the isle of Great Britain beginning with the Romans and continuing until the Norman invasion. You invade, you fight off invaders, you fight with neighbors, you have tons of fun. In a lot of games which try to convey a sweep of history this period is part of a lull between the glories of the ancient world and the dawn of the Renaissance.
Instead of being the "boring turn", Britannia is a joy to play.
One last game to round out the Early Middle Ages: Byzantium by Martin Wallace. It seems like something from right field (to me at least). We haven't had the east on a game board since Pax Romana; why go back? For me this game is a 4 player short hand for a lot of history which doesn't seem to be covered by 4 player games.
It covers questions like: what happened to the eastern empire, how about the crusades, what are the origins of the Renaissance, and so on. The short answer to the question of the eastern empire's fate is that Islam rose. The short version of the crusades is that the Arabs won Jerusalem from the Byzantines and then the Christians tried to retake it. The cliff notes origin of the Renaissance notes that Greek scholars fleeing from the collapse of the Byzantine Empire reintroduced classical learning to Western Europe.
Now, admittedly, there's more to it than that, but we can cover that in part 2 (which includes the Renaissance). See you then.
If there are any great games that I omitted or you disagree with the inclusion of one of the games then feel free to send a comment.