I agree with hanging it up to dry, but only after you've given it a rinse with clear water, I think soapy residue makes mildew worse. Some mats are better than others and the best I've found are the ones from Sainsburys for £5.
What I do to reduce mildew, is after my daily shower, turn the mat over and give it a good blast with the shower head, then a quick spray with a anti bacterial bathroom cleaner, slap it on the wall tiles to let it drain down into the bath.
It only takes a couple of minutes and it keeps it pretty clean.
Apparently, soaking them in sterilising fluid, like you'd use on baby bottles, will get rid of mildew on shower curtains, so I imagine it would work on mats too. Milton is good.