My host doesn’t support the graphics library that Movable Type prefers for making thumbnail images. Six Apart shows you how to use NetPBM, which is supported.

I also discovered that the version I’m running doesn’t do PNG graphic handling. That’s easily fixed. In lib/MT/Image.pm, change the line:
my %Types = (jpg => 'jpeg', gif => 'gif');
to:
my %Types = (jpg => 'jpeg', gif => 'gif', png => 'png');

As a Perl guy, I’m mildly horrified with the way this code works, but it won’t bite most people.