Since neck straps are more or less disposable most people don't mind if they break. Knock-offs can be had on ebay for a few dollars. The stock cannon ones last ages (I've had one for more than 5 years now and no issue aside from cosmetics).

If you want to go the way of a 3rd party neck strap i reccomend the unfortunately named 'R-strap' from black-rapid. See here. It makes a 5D with 70-200mm feel like its half the weight!

However, if your super keen on your strap, the following is how i clean my climbing rope :P So you it should work wonders for your neck-strap as well. (Rope needs to be maintained so the rope doesn't fail when you fall)

To clean your neck strap wash occasionally by hand in warm water with a mild soap, rinse free of the soap, and then hung up to dry in the air. Avoid direct sunlight, do not use a dryer, and do not place the neck strap above a heat source. If you can, the strap should be stored, preferably after drying, at room temperature, ideally in a storage bag.

The only thing i don't do with my straps is take them caving/crawling through mud. I've got a couple of pieces of climbing tape/rope for that if i need them.

Hope that helps!