Cleaning Milanese Loop band for Apple Watch
I'll expand on Apple's sparse support article.


Remove the band from Watch.
Take a clean cloth (I used a microfiber cloth) and dampen it with water.
Wipe down the band, stroking it gently and repeatedly. 
Turn the band over to the other side and repeat step 3.


The grime should be gone from your band and deposited upon your cloth, as illustrated below.