I've used 2 so far and have had good experiences with both. Not sure if you are looking for enclosed or open transport, but here are my recommendations:
DAS Auto - Open trailer transport like you normally see on the freeway, great customer service, delivered ahead of schedule and the car hadn't been driven or soiled in any way (which are common complaints that I hear). Meridian --> San Diego cost about $1000, door to depot.
Concours Auto Transport - Enclosed carrier, they deal with a lot of high end exotics and collectors cars. They moved my project car (that wasn't driveable at the time), and did a great job with it. Oregon to Atlanta ran about $2000, door to door. My only complaint was the timeframe it took for the move, but they are a fairly small specialty carrier, so they really only have 1 or 2 trucks moving and they try to fill the trucks (read: extra stops en route) before they make the trip.
To answer some of the questions lumpy posed, both of these companies allowed us to store stuff in the car; DAS wanted items in the trunk only, while Concours had no limit (I shipped a whole boatload of car parts I needed for the completion of my project). Having them deliver or pick up at a central facility/depot can save you some money, but of course you have to think about convenience for that one. The depot in SD was about 45 mins from my house up in Oceanside. Also, neither company had a problem letting an authorized release agent sign off on the car (my dad for the OR->GA move, and a friend from Meridian for the MS->CA move); you just need to make sure they have contact info for that person.
I'd highly recommend either of these carriers depending on what kind of transport you are looking for.