Round 1: There's just so many ways the Gotei could win here it's kind of ridiculous.
Round 2: Gotei takes it again but has an initial disadvantage against Itachi's genjutsu's. However, with enough rieatsu the majority of the captains are able to break through the illusions by flairing their rieatsu much like the shinobi can flair their chakra to release themselves from genjutsu. The majority of the Akatsuki are defeated easily. A few of the captains may be defeated by Itachi, Pain, or Tobi. But in the end the combined efforts of the captain's superior fighting abilities pull them through.
Round 3: Yamamoto takes it. He could solo everyone there. His bankai solves every problem he could encounter from any of them.
It may sound as though I am simply fanboying in Bleach's favor but I have actually given this matter significant thought. Unless we are including Madara with the Akatsuki there is no one who can match up to Yamamoto on the Akatsuki's team. Madara is the only one with a chance of possibly beating him and I still don't even think Madara could beat Yama in bankai. Besides Itachi's genjutsu, the Captains working as a team could effectively counter, overpower, or completely negate most of what the Akatsuki could do to them.
Yamamoto can solo the entire group himself if he wanted.
Ukitake can redirect any jutsu they have back at them.
Hitsugaya can freeze anything and everything Kisame throws at them or Yamamoto's bankai just evaporates all the water with his bankai.
Kenpachi can cut through any meteorites Pain throws at them with his shikai. In all honesty, the only one I could see killing him is Hidan and that's only if he was lucky enough to get close enough to Kenpachi without being immediate cut completely in half and if after that he somehow managed to get Kenpachi's blood (most likely due to Kenpachi's brash nature) and performed his ritual. Otherwise Kenpachi can tank anything the Akatsuki can throw at him except perhaps maybe Amaterasu.
Unless Obito is phasing every part of his body when it happens, Kyoraku can catch him by surprise and possibly kill him with his shadow stab ability Kageoni. Alternatively, If Sui-Feng catches him by surprise gets two hits on him in the same spot with her shikai he's done. Also alternatively, Obito attempts to grab Yama while he is in bankai to transport him to the other dimension. Obito bursts into flames when he touches him. Yama has the heat of the goddamn sun enveloping his body.
Byakuya can handle Itachi's Susano'o the same way Gaara handled Madara's by bringing his bankai blades up from under Itachi's feet and killing him from inside his susano'o. Factor in Byakuya's ridiculous speed and kido mastery and he could probably take on the majority of Akatsuki himself one on one as well.
Mayuri could simply go bankai and poison them all and wait for the poison to do them in or simply weaken them enough for the rest of the captains to easily finish them.
Shinji could use his shikai to confuse them all with his inverted world ability, allowing all of the Gotei to gain the upper hand and clean up easily. Even with Itachi's Sharingan he's still fucked up due to the illusion effecting him through the shikai's initial mist. The mist is absorbed through the lungs and goes on to mind fuck you. It's not so much your eyes play tricks on you as it is your brain telling you the wrong information to process. Even Itachi's eyes can't see through this if his brain is what's fucking with his vision and telling him the wrong ways to react.
Sui-Feng is fast enough to simply kill Deidra with her shikai before he even has a chance to use his bombs (microscopic or otherwise) or she could just blow him the fuck away with her Bankai nuke.
Kensei could wipe out Kakuzu with one punch from his bankai. Reminder: Kensei's bankai, as long as his fist has contact with his enemy, sends constant vicious explosions coursing through the victims body until the fist is no longer in contact with them. In short, Kakazu's five hearts would be destroyed in a few seconds.
Kommamura's Bankai could easily dispatch Sasori's 100 puppet army. Kommamura is also bulky enough to tank anything the third Kazekage puppet's Iron sand can throw at him. Add into this his human form that is effectively immortal and Kommura could potentially tank anything the Akatsuki could throw at him long enough to kill most of them. His bankai could also take Itachi's susano'o in a straight fight. It's shield may be impenetrable but the Susano'o itself would probably crack and break after awhile under the bankai's superior physical strength and weight.
Unohana play the support role and can stay back and heal anyone who's been injured thus giving the Gotei the advantage of a healer. Or she can take on Hidan and destroy him easily without him ever being able to touch her due to her mastering of every damn sword technique EVER and having incredibly fast reflexes.
To counter their genjutsu, Rose can use his own illusions with his bankai.
and don't even get me started if we were counting former captains in this roster for the Gotei like: Aizen, Urahara, Gin, Yoruichi, Isshin, Tousen, and Love. The top members of Akatsuki could put up a good fight but in the end the Gotei would win this hands down.