For JFK, I have changed recently from NYC yellow cabs to Cloverdale (Big Apple) Car (718-236-1000). They can meet you in terminal but since they always have a car or two out at the airport, if you call them as your foot hits the ground, they will have a car for you by the time you walk to the door of the terminal.
Since I (and most other NYers) only use JFK for international travel, their meeting point at the new International Arrivals Building is across the first street, on the island walking to the right near the yellow cab line.
My recent change is due to the medallion taxis being so uncomfortable for anything more than a few minute's ride. In the Fords (which are now virtually the only cars that are used), the way the seats are situated and the shield behind the driver (in front of the passenger) makes you feel as if you are sitting in a hole and has made this one of the most unattractive modes of transportation on the planet (possibly surpassed only by the NYC subway).