Region, hands down.
The RTW ticket will allow you to city hop, but you'll end up disoriented & feeling unconnected. After a few weeks you'll wonder what the point of it all is.
You'll feel the same way anywhere 3 mos on the road, but if you stick to one region a few things start happening:
+You run into travelers who are on a similar route, and who share the same experiences. That will not happen if you pick world cities at random.
+You start to pick up more of the language (a lot, if you try hard) and start to blend in just a bit more. You start to feel a connection to a place. An ownership, almost. I spent four months in Indonesia a decade and a half ago, and I still feel a connection to the place that I would not have if I had just passed through.
- You start having deeper relationships with locals. You are not always thinking about how much time you have, or where you have to be to catch the next plane, so there is a lot more room for spontaneity.
+You start to discover what the meaning of it all is. Not that you won't loose it again, mind you.
Your regions are all so different! My biased opinions: Save the Pacific for regular vacations (it's home to me, but it is expensive to travel in!). SE Asia is classic, an awesome intro to world travel & has thousands of options (from party islands to remote villages to Lonely Planet vortices. Indonesia or Turkey could both occupy 3 months just on their own. Greece might be pricey - I've only been to Mykonos and Samos. Egypt is rough for the backpacker. Go there, see what you want, and get out. If you are in the area, though, Jordan is super friendly, has Petra, Wadi Rum, Crusader Castles, Roman ruins, deserts and valleys and the Dead Sea, & is still just a bit off the beaten path, though I can't imagine it staying 'undiscovered' by the masses for long. They say Syria is even more cool.
3 potential hella-cool life-changing 3-month journeys: Cairo to Istanbul overland, Bangkok to Bali overland, or use Bangkok as a base to loop around SE Asia.