We arrived in Term C from JetBlue. We rented from
Alamo. They have a kiosk there where you can check in and get all your papers. You either have a choice of walking (exactly like carlbarry said) or going outside and getting a shuttle that will take you to the transportation area of Terminal A.
The shuttle took a while to come, then they have to put everyone's luggage in the compartments, then wait for the shuttle to fill up, then it has to drive all the way around the airport while fighting traffic until it unloads at Terminal A.
There you can pick up your car. The shuttle process took about half an hour as well. So it doesn't save any time, but it's very little walking.
We had carry-ons, so it took us a little over an hour from the time we got off the plane until we drive out of the airport in our rental. If we had checked luggage to claim, I'd add another 30-45 minutes.
You can drop your car off at the Terminal C car return. It's a huge parking garage that's not being utilized yet.
Yes, it's a huge inconvenience!