Rewards.... as far as i can see there are none .  Which, y'know is fine i guess.... more often than not the story is a reward in itself.  In MW2 that's debatable.
I felt the Museum was a let down, i was hoping for "Arcade Mode" again or something. 

I have seen though, as with the first game, when i look at players profiles online, nobody has any Single Player achievements.  Hardly anyone is playing it.
So i sugest for MW3, rather than just suggesting or highly recommending i play Campaign first when accessing Multi-player, reward 1 or a couple of novel, if not game changing multiplayer awards.
Examples: 
10 Characters unlocked  for Clan tag than just 4 - a saved the world badge of honor.
Call-signs unlocked for level completion, that maybe grant small bonus' such as 10% extra sprint time, Noobtubers glow red or makes your man shout "I'm OSCAR MIKE!" when you sprint. 
Completing on Veteran could give you a good perk like "Knife Proof" or 2 Claymores.
 
Suggestions?.... what would you like?.
Also I'm sure we've seen this thread all before but it gives me an excuse to say my ideas.

I have no clue why people would play only multiplayer and never touch the single player. That's just silly.

I wouldn't mind seeing new skins or other aesthetic rewards for completing the single-player campaign, but I don't think there should be anything functionally game-changing.  People shouldn't be penalized if they don't enjoy the single player, so new perks or gameplay bonuses would be a negative.  
 
I still root for the idea that in World at War, when you use the flamethrower for a kill, the Mortal Kombat guy should pop up and shrill, "Toasty!"

" I have no clue why people would play only multiplayer and never touch the single player. That's just silly. "