It depends. If it is a dungeon I've already been to, then I'll breeze right through it. If it is a dungeon I have not discovered before, then I might wait for a quest for it or just say,"What the heck, I'll explore it."
For me playing rpgs is all about the dungeons with unusual fascinating architecture full of monsters and glorious treasure so i never wait for a quest to pop up. If i have to do the dungeon over its no big deal and usually things are different in some way like higher level creature spawns etc to enjoy.
yes, when i found out some of the dungeons had random quests in them i started clearing out any new ones i found. Some of them reward you with new followers, and i'm all about getting new pack mules