I like the earlier suggestion of having a player resolve his or her Upkeep, Movement and Encounter phases, before moving onto the next player, and resolving Mythos after everyone's done.
It could seem a good solution, but it alters *a lot* everything in the game. Just imagine you need to enter a gate. During the "normal" Upkeep phase, you select an investigator to clean the streets, another one to do other things and the last one to dive into the gate. You set the skill sliders for all investigators accordingly. Then, the first investigator tries to clean the road, but fails and it's knocked unconscious. Now you don't have any other investigator ready to slay the monster so that your last investigator can enter the gate safe. Now imagine if everyone plays the whole turn: investigator A is smashed. Oh, too bad, let's try with investigator 2. And so on ::faceplam::
Plus, it could screw also your plans: investigator A returns to Arkham, but has not enough clues to seal. So he skips his Encounter Phase. Investigator B passes by, and tosses him an Elder Sign. In the next turn, investigator A seals with the ES (with the risk of losing the game if the doom track is almost full, or anyway with the risk of a monster surge and so on). If played according to the rules:
- movement: investigator A returns to Arkham, investigator B passes by and tosses him the ES
- encounters in Arkham: investigator A seals the gate with the ES
I could add several more examples, but I guess the point is clear: Phases were studied in order to allow a greater interaction among the investigators (the game is coop, so these dynamics are vital to enjoy the Arkham experience at its best) and to allow a greater general strategy.
The game could seem very long, but once you get accustomed to the rules, you should be able to play a game in 90 minutes or so. Maybe a little more, maybe a little less, but still, unless you don't go for the score, it's very rare that a game lasts for more than two hours