Well, I had to get pretty aggressive, but I got it. Here's what I did:
1. Heat all edges with a heat gun at 180c (356f) for one minute.
2. Use a suction cup to pull on the screen. I also plugged a usb cable into the bottom of the phone to give me another place to carefully apply counter pressure.
3. Slowly wedge a razor blade between the screen and the body at the bottom of the phone and work it until I was able to get a plastic pick in.
Eventually I was able to use the razor blade to clear enough adhesive that I could slide a plastic pick in. I did have to carefully use the razor blade as a pry - sliding it in a bit and then putting gentle pressure on it to try and pry the screen up. The glass actually started breaking a little where I was doing it. Then I just had to slowly go around the body to loosen the adhesive.
Obviously this approach will only work if you know you're replacing the screen, because you're almost guaranteed to damage the screen you're removing.