The ancient Hellenes kept a lot of animals--mostly farm animals like cows, oxen, pigs and chickens--but also dogs for hunting and probably also cats, weasels, ferrets, birds of prey and a variety of other animals. In ancient Hellas, animals had functions to fulfil and weren't just solely there for company. They weren't pets like we have them today and they didn't usually get buried with rites. Dogs (and very seldom other animals) did sometimes get buried with grave markers, though, so they definitely did care!

Every once in a while, I get the question how to do a ritual for the death of a pet. The most common animals being commemorated are dogs and cats, so I'm going to tell you a bit about those two today.
A while ago, I posted a basic outline for a rite of passage on my blog. I would adapt this rite to include the Gods mentioned in the text below. You can find the rite here.

Dogs had a very special and particular place in ancient Hellenic society. The Hellenic word for 'dog' is 'kuón' (κύων), and they were used as guards and in hunts for game. Dogs used for the hunt had to have very different qualities than those needed to guard a house, herd, field or person. For hunting dogs, swiftness was paramount and for guard and shepherding dogs, Marcus Varro, a notable Roman scholar and writer, advised the following: a large head, drooping ears, thick shoulders and neck, wide paws, thick tail, a deep bark, and be white in color so as to be more easily recognized in the dark, were important.

Artemis is associated with a lot of dog-related mythology. As the Goddess of the hunt, she is often depicted amongst her own hunting pack. Another Goddess associated with dogs is Hekate. She is often described as being of dog-shape or tended to by dog. The dog was also her favored sacrificial animal and the meat of the dog was eaten solemnly. That it was eaten at all is a testament to Hekate's rule extending even beyond the Underworld, as Khthonic (Underworld) Deities often received their sacrificial offerings in a holókaustos. In the rite, Hekate should replace Hermes and Artemis should be added before Persephone.

Modern Athens is home to countless feral cats, however, the status of the cat in ancient Hellas is unclear, as images of cats are fairly rare. Cats were probably kept by Hellenic households as liminal animals--animals who were free to come and go as they pleased, roaming houses and streets alike. This may account for the fact that few cat bones have been found in domestic situations. Historically, the main reason to keep cats is to get rid of mice, only in ancient Hellas the job of 'mouser' seems to have been handled by weasels and ferrets, many of which were also considered liminal pets. Yet some clear representations of cats do exist, primarily on funeral stele. These show cats on leashes, or being shown off as prized possessions, so we do know that some cats were updated to the status of 'pet'.

There isn't a God or Goddess of cats, but there is a Goddess associated with cats: Hekate. There is one single piece of mythology I have to base this on: the myth of Galinthias, the nurse of Alkmene, transformed by the angry Eileithyia, but received by Hekate as Her animal. Dionysos is the God of wild, predatory, felines--of those felines who pose a wild threat, who can rip flesh and inspire fear. He might look out for the smaller felines as well. In the ritual, Hekate should replace Hermes and Dionysos should be added before Persephone.