Dogs do actually have sweat glands on the bottom of their feet, but yes, they mainly cool down by hyperventilating (panting). The rapid fresh air sucked and blown over the many blood vessels in the tongue cool the blood.
A more effective cooling method is perspiring (sweating) where body fluid is forced up through the skin pores and cooled by the air. It is a complicated way of transferring the cooling effect to the body but effective because of the large area exposed to the air. This method however is a heavy drain on the body fluids and is employed by animals which, in their natural habitat, are able to replace this by drinking. Animals with heavy fur or feathers where sweating would not be practical have other ways of cooling - panting being only one.
Many animals have glands on the lower parts of their legs [including cats] which they lick or urinate over to keep cool. (you didn't really want to know that, did you?).
Elephants don't have very large ears because they were naughty as calfs, the ears have many blood vessels close to the surface which has a cooling effect on the blood flowing though them.