Hi there... I brought my 60lb dog on American, he went with baggage (I believe Continental will ship them too, maybe the word "cargo" threw them off, you may want to call them and ask about a dog specifically). American had the least restrictions; recent shots and an authorization to travel outside the us from a vet (vets are familiar with these) dated within 30 days of the travel. He had to be able to stand and turn around in his kennel and I had to provide instructions for food/water (I suppose in case the flight was grounded for many hours). I was not allowed to sedate him (although my vet gave me sedatives and I did). I was terrified for him to do the flight and it actually was much easier than I imagined. Aruban customs did not ask me for anything, no forms, nothing, they just gave me my dog when I landed (I can't promise that's the norm). I paid American $150 (similiar to an oversized extra bag charge) and they were pretty accomodating. I did get different stories from all the airlines every time I called, so I would recommend calling a few times just to make sure all your papers are in order. The only other possible issue (although I can't imagine it happening) was that only 3 dogs were allowed in cargo per flight and I could not reserve a spot for him, so had I shown up and there were already 3 dogs on the flight I would have been SOL.
I believe Continental had a dog specific customer service area, but I may be thinking of another airline.
Good luck and if you love dogs, be prepared to adopt a few here, I've already taken 3 more from the streets.