i have a Logitech g600 and the braided cable got really dirty.
here is a simple fix.
first off do not try pulling he dust off as it will rip off some of the braiding with it.
just get a fairly damp, or moist cloth and wipe it a couple time, apply a firm pressure and its clean.
don't use a microfiber cloth , its not rough enough, or tissue, the tissue rips off and it gets all messy.
i used an old shirt, good luck.