How to clean hard-to-clean vases

If you've ever tried to clean a vase with a narrow opening at the top you know it's not actually that easy, since most hands/sponges/brushes are all too big to fit through the hole to scrub down the interior. My favorite vessel for displaying fresh flowers has a bulbous base that culminates in a skinny neck, and lately I've been noticing a film of mildew-y plant residue stuck to the bottom of the vase. I tried rinsing it several times with soapy water to no avail, and had all but given up until I read about the following trick in the comments section of apartmenttherapy.com:

Take a couple ice cubes and throw them in the vase along with some coarse salt (I used about a tablespoon of kosher salt). Swirl it around for a minute or two, then rinse. I just tried it and it worked brilliantly. The same commenter also wrote that the tip actually originated from an old diner waitress, who said the technique is great for getting the stains out of coffee pots that have been sitting on burners all day too.