First off, start thinking fresh. Forget all this nonsense about a double story line or using the old characters. Sonny wasn't alive long enough for there to be anything interesting about him and Micheal died at the end of 3 and that's how most of us interpreted it. Let the old trilogy RIP.
Don't call it 'the godfather' either...come up with something new that's relevant to the storyline at hand.
I want to see Vincent (ANDY GARCIA OF COURSE!!) as the main character, first having to defend his new title of Boss against the other four families and detractors in his own family who want to be boss themselves. There was plenty of (real) mob activity in the 70's and 80's to build a new fresh story out of. EDIT: Not to mention the dawn of surveillance technology which wasn't really touched in the first 3 films.
I love the idea that most of you have about Vincent returning the family to the 'street crime' empire. It should make for some good violence. Perhaps one of his challenges is that the family lost most of its money in the Imobilliare debockle and he's really starting from scratch. Perhaps the Corleones are poor now?
Some say Vincent shouldn't be successful as boss. I disagree because the underlying story here is going to be 'how successful would Sonny have been as boss with his temper?' The evolution of Vincent's character is going to answer this question.
Finally, I would film it similar to the Sopranos to give it a modern feel. I would even bring some of those guys in. Imagine Paulie in a GF film!! Sweeeet.
Ah, just my 2 cents.

I always thought GF3 should have been a street level action film in which Solozzo's children came back after Michael. No reason they couldn't come after Vincent.