It sounds appealing (and it's definitely cheaper than general anesthesia), but there are a number of reasons why veterinarians- and veterinary dental specialists in particular- strongly advise against it.
Here's the American Veterinary Dental College's official source for information regarding anesthesia-free pet dental cleanings. There's a ton of great information regarding veterinary dentistry in general, including some thorough discussion of the pros/cons of anesthesia-free dental cleanings.
Lest you think I am just going to toe the party line on this matter, here's an article from the American Veterinary Medical Association ("Below the surface of anesthesia-free dentistry") that does a good job explaining veterinary medicine's internal debate on the topic.
SPOILER ALERT: Everyone still says you should get your pet's teeth cleaned at a veterinary hospital under general anesthesia.