without using a carpet cleaner
For the heavily soiled areas, use a rug stain remover, like Spot Shot or Resolve carpet cleaner. For the rest of the rug, use a cloth dampened with a white vinegar and water mix. It will clean and deodorize the rug. Turn a fan on, directed at the rug to help speed the dry time.
Here is a site that I would highly recommend. They have very high quality stain removal techniques. I would suggest repeating the process after the first application, and rinsing the rug out with cool water in between applications. Best of luck!
