i either do what Jason does, ie nothing (90% of the time)
or
I do what eddie does; use each smaller section as a pushrod to push a small patch of paper towel through the section as well as dry the exteriors. Generally speaking, I just put a single rectangle layer over the end of the section thats one size smaller than the one I want to clean.
That way it just very lightly touches the insides of the surface.
I also do one other thing that I've been doing for years on my western rods.
Get some lemon pledge, or any waxy furniture polish. I prefer lemon just cause i like the smell, though Ive become partial to the rosemary mint my wife has recently switched to

Spray some on a cloth, and "polish" each section of rod. Gets them really nice and clean and smooth feeling. Buff it out so you have a minimum amount of residue on the rod. If you do it right, the light waxy coating will actually help keep the rod clean rather than attract grit.
(if anyone is into cycling, it just like those wax chain lubes that keep the chain cleaner by shedding dirt\grit\etc)
Water however is the least of my concerns....its the GRIT I'm worried about.
Once i didnt check my rod after getting it wet and the next time I pulled the rod out, I couldnt extend the rod because of some grit that was in there.
Whatever you do, DONT EVEN TRY to extend a rod with grit in it if it doesnt want to freely extend. You WILL score sections of your rod.