Thanks to all for the replies - looks like I have some good ideas to work with. One thing that I am certain will not work is water. This is a basement floor, and the carpet has been wet from footing/basement drains backing up several times. This last time it was very bad, and the decision was made to get rid of the carpet. Much of the carpet was wet when we took it up, and water was squishing up out of a lot of the old foam as we scraped it up. This glue held tight after having been soaked with water for a few days.
I will look into the various removers/strippers first. There's enough of this stuff on the floor that it is hard for me to imagine that just taking a grinding wheel to it would not quickly result in a hopelessly clogged up wheel, which would get real expensive in a hurry. Unfortunately, it is also a relatively large area, probably on the order of 30' x 60'.