I like it when the main plot progresses even if my PC is not directly involved. That gives a sense of being in a living world ... where Narlen Darkwalk and Gorpel Hind and Drizzt are out there doing their own things whilst my PC is trying to gain enough experience to be able to go after the main objective. I would love to see recurring random appearances by familiar characters in BG III. In fact, you could set them up with a sort of "Mad Lib" set of dialogs, where certain dialog words are randomly determined from a list. "Whoa, I just got back from a (dragon) lair and you'll never guess what we found! It was a (ring) once possessed by the late (queen) of (etc.)..."
And then when the main plot requires action from my PC, there are many ways to deliver the notice. BG I used the dream sequences quite a bit, and BG II had a lot of messengers running around Amn trying to find me. And of course sometimes Elminster would show up with advice. Those are all great devices to pull the PC back into the main storyline.
