Honestly, I would NOT use fabric with an odor for dog beds or cat beds. Dogs have a sense of smell something on the order of 10,000 to 100,000 times stronger than humans. Cats are estimated to be about 14 times more sensitive to smells than humans. Even dogs and cats care about what they sleep on!
I still think the best use for fabric in which odor is not completely removed is rag rugs. They can be used outside as well as inside the house and still serve a purpose.