First off, I hate action movies. I hate them because they tend not to have much of a plot that I can get into, characters I can't get into, and I find the action dumb.
But The Bourne Series
manages to address every single one of those criticisms. Heck, while it clearly has a lot of action, it's better described as a thriller.
The plot gets going and just doesn't stop. Bourne's former bosses/allies want him dead, and they're willing to pursue him to the ends of the earth to achieve that goal. As for Bourne, what he wants is simply information about who exactly he is and what is going on.
What follows is a cat-and-mouse game as both Bourne and his enemies try to outsmart each other left and right. For example: when Bourne is in Germany and gets cornered in a house, he calls the police and says he heard gunshots and believes Americans were involved, then promptly fires his gun a few times to attract attention. Bourne escapes just in time as the German police show up to arrest his pursuers. Earlier in that same scene, Bourne leaves a fan running in a bathroom to lure one of his pursuers in; he knocks the person out as soon as he's able.
There's tons of brilliant moments of Bourne thinking on his feet and improvising solutions to distract his pursuers or escape them. In fact, he mostly does that and engages in very little actual combat, and when he does fight, it's mostly non-fatal combat. This man isn't out to kill, only to survive. That makes him very different from other action heroes, and someone I find more relatable despite his large amount of knowledge and specialized training.
Another thing I really like is Bourne's relationship to the temporary allies he gains in his missions. He meets a woman in the first movie who seems to have a shady past, and quickly enslists her help. The two work together, carrying out Bourne's plans, and she proves to be a very useful ally and a neat character, creating a fun dynamic. In the third movie, Bourne discovers a news reporter who's being pursued and attempts to help the man escape, giving him exact instructions that he follows... until he freaks out and loses it (and pays the price).
Basically, The Bourne Series
is the opposite of what I hate about action movies - it's intelligent, it has characters I like, it has action I actually find compelling, and a good story.