Airing is a wonderful solution that I'm using more and more. Just yesterday, we draped the kids' down duvets over the deck railings for an afternoon of sunshine. The breeze moved the air through the duvets and the sunlight disinfected them, freshening them beautifully. They weren't soiled or dirty because we use duvet covers, but the down certainly benefitted from an afternoon outside in the fresh air. Both quilts smelled freshly laundered once we brought them in.
I've tried the same technique with clothing that isn't soiled. A garment that isn't dirty or sweaty can easily be freshened if it's hung outside for a few hours in the fresh air and sunshine. I use a drying rack that I keep by the front door - I can easily take it out to the front verandah and hang something on it in mere seconds. The railings on our front verandah work well for larger items and will have to do until the snow has melted enough to reveal the hole for the rotary clothesline!
Drapes can be shaken to release dust and hung outside to be freshened, mats and rugs can be hung over a railing, coats can be spot cleaned and hung outside, stuffed animals can be swatted to release dust then pinned to a rack in the sun, toss cushions can also be swatted to release dust then laid out on a drying rack...
|toss cushions from the living room|
Lest you think I've lost my mind (and don't do laundry anymore), I'm particularity picky about our linens being clean and any garment that is dirty IS promptly washed. The effectiveness of airing is astounding though - harness that sun and fresh air to aid you in your home keeping. This technique saves water, electricity and soap but effectively freshens many things that don't really need a wash...
Give it a try - your Grandma would be very pleased (she knew best!) :)