No, as others have said, you cannot use the Run action when there is no straight path available.
However, you're mistaken if you describe the alternative as "walking". The average D&D human using a double move action covers 60 feet in 6 seconds. That's a speed of 6.8 mi/h, and moving at that speed is not walking! It's closer to a brisk jog -- moving as rapidly as you can while still maintaining control. This is even called out in the rules for tactical movement:
Use tactical movement for combat. Characters generally don’t walk during combat—-they hustle or run. A character who moves his or her speed and takes some action is hustling for about half the round and doing something else the other half.
The Run action is sprinting full out at your max speed, which is realistically something most people could not do within the twisty corridors of a dungeon. If you tried you wouldn't get as far as more careful movement, since you'd have to stop at each corner or trip over yourself. The rules here actually make a lot of sense.
One final note: there is a feat, Fleet of Foot, that lets you make a single turn while using the Run action. I vaguely recall a class ability or second feat that improves this, though I can't find it now.