Tony:
With Tony, he rarely gets annoyed and when he does, he stays in his lab and tinkers with his suit.
Steve:
Steve gets mopey when he's annoyed. He just walks around the tower, sulking. You eventually cheer him up.
Thor:
When he's annoyed, a giant storm brews and he just sits in his room, listening to the heavy rain and thunder.
Bruce:
Well, you know what happens when Bruce gets angry.
Clint:
Clint goes to the archery range and shoots until he feels like he's going to collapse from sleep deprivation.
Bucky:
Bucky works out and spars so he's distracted from whatever pissed him off.
Peter P:
Peter swings around NY and takes walks to calm down before returning home.

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Avengers Preferences and Imagines
ActionTony, Steve, Thor, Bruce, Clint, Bucky, and Peter P. I might include Loki and Natasha in some, comment for other characters you want. :)