The farther they get away from the characters of DA II the better. DA II made the mistake of underestimating it's predecessor; All the characters they introduced in the first game are either gone or reduced to cameos, and the story doesn't take advantage of the momentum or dangling plot threads of the first game. It functioned more as a spin-off and was written like a fan fiction.
The first thing DA III needs is to cement itself back into Ferelden and with the characters of the first game.
The next thing it needs to do is to stop trying too hard to be funny. Ever notice that even Mass Effect 3, arguably one of the bleakest games Bioware has ever produced, is still funny? Why? because humor is integral to how humans interact, you don't need to force it.
Lots of conversations in Origins are funny, but almost none of them are there for the sake of humor alone, the humor flows from the characters and conversations naturally, but the game never forgets that the scene is still there to learn about the character.
In DA II, the game went so far as to dedicate an entire third of all your reactions (A whole three choices!) to shits and giggles. It seems that DA II remembered that Origins made people laugh but forgot about everything else. Most of the characters can be summarized in one phrase, and that's all you ever learn of them.
Fenris: Bitter and hateful (And he's actually one of the better ones, at least he has motivation)
Anders: Strong sense of justice... oh wait, that's the character they attached because he's so fucking boring.
Isabella: Fear of commitment
Merril: Naive
Sebastian: Religious
Avaline: Honorable
Varic: "Witty"
Hawkes family is completely nondescript
Dragon Age II kicks off emotionally forced, trying to make you feel bad for people you don't know during an event you've already seen the conclusion of. It completely ignored what Origins had set up for it.
However, DA III should not make the mistake that it's predecessor did, and ignore the things hat DA II got right, or rather tried to get right.
Now while I thought that the person who decided that DA II should look like a Medevil Dragon Ball Z should be drawn and quartered (I just blew someone up with a knife?), I think some of the mechanics for the combat are better. The feature that works best in DA II are the cross class combos, they're great and should be kept around.
I also like the concept of the skill web. In practice, it just allowed different skill trees to be structured differently (Which is good), but the concept of being able to customize a character down to every individual ability is an intriguing one, but I have my doubts that such a radical system will make it into the game.
Not everything has to be about the end of the world. The disjointed plot of DA II made me worry that Bioware have gotten much to used to this plot crutch, especially when (Most of) Biowares characters are good enough to motivate you to action all by themselves. You don't need an all purpose motivator to get us to want to spend time with these guys. DA II's focus on character wasn't a bad thing, the characters in question just sucked. The story of one group of people getting by and dealing with political issues could be great. That's two thirds of the overarching plot of A Song of Ice and Fire, arguably one of the reasons Dragon Age even exists.
I also thought the overarching plot(s) of DA II were pretty good. The Qunari make for great villains, because even though their just people like us, their way of life is incompatible with our own, and it's hard to fault them for being different.
The conflict between Mages and Templars is an inherently intriguing one, because like the Qunari issue, both sides have valid points, and while DA II played heavily in favor of the Mages, the idea is still fresh.
But this also presents a problem; Both of these conflicts could have easily been entire games on their own, but instead they were both half-assed in one game, so that means that one half of each idea is already over and done with, so the ideas can either be artificially stretched to make for two whole stories, or half-assed again. I think the best thing to do would be to just abandon the trilogy model and make two games. I'd rather the ideas be padded at the start and end up well than just debunked or poorly done
Dragon Age could be great again, it's just a matter of learning from it's mistakes.
EDIT: I heard somewhere that, again, the player will be playing a new protagonist in DA III. I'm not sure that's a good idea, as I said, one of the biggest issues with DA II was how little it involved itself with the characters from Origins, and if we're again playing an entirely new character, I'm concerned that will happen again. Even if they do appear, I'm sure I'll regularly find myself wondering what became of their relationship with the Warden. I don't know why Bioware are so resistant to playing as the Warden, silence is a small price to pay for story momentum. I hope they're not disrespectful enough to kill him/her off screen.