Swinging a cane is never a good idea unless you intend to strike someone with it. Walk as you normally would while allowing the cane to hang from your hand, and "plant" the cane as you step with the opposite foot (i.e., if you're holding the cane in your right hand, you place the rubber tip on the ground as you step with your left foot). For a more thorough explanation, read this page on the Fashionable Canes website.
Ive usually heard this as the way to walk with a cane.
However, having been in a couple of motorcycle, er, 'altercations' , I have found that keeping the cane on the same side as the injured appendage works much better for me. I can take the weight off the foot/ankle as I step down on it by having the cane on the same side.
Not that my way is the 'right' way. Is just what works for me so I offer it up here.