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I wipe the razor down with a towel after every use - between blade changes I'll give all the pieces a good wipe (I generally use 2 or 3 piece razors) with a towel and nothing more. If I do happen to be using a TTO, I'll give the inside of a head a light scrub with an old toothbrush.
I really don't see the need to use chemical cleaners on a regular basis - its just soap residue (for me anyhow - I'm assuming that's the case for others as well...).
I've been wiping the residue off with a couple Q-tips, but now that I know others leave the residue on for longer than a day (and therefore I have some peace of mind I'm not ruining my razor), I think I'll switch to Scrubbing Bubbles once per week.
I find that running the razor under really hot water for 20 seconds or so after each shave gets pretty well all the soap off. Every few months I put my razors in an ultrasonic cleaner with some washing up liquid and they come out looking like new.