The article's top solution? Prepare yourself.
Have the animals live somewhere else while you're relocating.
Yeah, we know. But think about it. It's not always practical or convenient to you to get them out of the house while it's being shown. It's not fair or healthy to your beloved pets to throw them in a basement, yard or other confined space during the showing, either (and, if they have fear or anxiety about the situation, it's not good for your house).
But would a family member or friend agree to take them on for a limited amount of time? Is it economically feasible to board them for an open-ended amount of time? It's something you should think about.
In the meantime, you can make some pet-related fixes to the house; waxing and polishing floors, cleaning baseboards and under furniture (even the least-sheddy of pets leave some trace), and getting rid of odors and stains (think around doors, litter boxes, beds and food bowls).
This won't be the most popular solution but, as the article says, if you want top dollar for your house, you'll at least consider it.
Consider that the nuclear option, but clearly there are other ways. If you're able to keep up on having the house clean and eliminating evidence of the animals, you can always have a trusted neighbor come get them and take them for a walk ahead of the showing. At the bare minimum, according to the article, you should put the animal in a size-appropriate crate and clearly mark it "Dog/Cat Inside: Please do not disturb."
Another practical tip, from the experts at Zillow.com Check your local laws and see if disclosing pet presence is required in your area. It often is. Some buyers will appreciate knowing because of allergy concerns. Best case scenario, you've done such a good job cleaning up after your pets, they'll be impressed by your housekeeping skills!
One other tip: Just because your house is spotless doesn't mean it's odorless. You may have lived with a particular pet's odor for so long that you simply don't notice it. Bring in a neighbor or honest friend and ask them to tell you if they smell your pets in your showing-ready house. If they still do, consider a professional cleaning of carpets and furniture. An article from HowStuffWorks states that undetected pet odor or damage can take up to $30,000 off the value of a home. Can you afford a deep-cleaning for that?