I take it you're looking for just straight up fighting? Something like Battlefield 3 or 4, CoD, several Free 2 Play online only shooters. Just stick to the multiplayer only. A few of those might have some 'establish the mission' cutscene, but that's just hiding the Loading screens.
Now, if you're looking for a game with a story to play, but with no cutscenes....uh, yeah those phased out in the early 2000's once they were able to make decent cutscenes.
I dunno. I tend to like the stories. Killzone, Max Payne, first couple Modern Warfares had 'decent' stories going on between the fighting stages.
Now, Quicktime events need to die in a fire. Totally immersion breaking for me when I have to suddenly play a Simon Says mini game in the middle of whatever I was doing a second ago.