Do good power cords have to be stiff?

Stiffness of a power cord has no effect on the sound. It only changes the mechanical characteristics and makes it easier to handle. As long as the current rating of a cable is not exceeded, the rest does not matter electrically & sonically.

Simple answer… NO!
 

I was using Nordost Vishnus in my mid level system. A clear improvement over stock.  But they too were somewhat stiff. Replaced them with Morrow MAP3s which sound just as good as the Nordost and are quite flexible. If I get the upgrade itch I will climb the Morrow ladder before going back to stiff power cords.

I've been using a Wireworld Aurora 7.  12 ga silver clad OFC.   Its flat design makes it pretty flexible.  I have it folded back with about an inch radius.  For my DACs etc, I agree with @thomaswv that Morrows are extremely flexible at least lower in the line.  

+1 on Audio Envy

Morrow Audio power cords are also flexible and easy to route