Follow up:

I finally gave up on having each header in its own band and put all the headers into the smallest group's band. Then for each field under the "Print when" checkbox, I set the print repeated values to "No".

In the "Also print" area:
I Checked the "In first whole band of new page/column" checkbox
Checked the "when this group changes" check box and set the popup to the appropriate group
Checked the "When detail overflows to new page/column" checkbox

And finally checked the "remove line if blank" check box at the bottom of the screen.

The report now looks great and the users will be much happier.

Anyway, just thought I'd jot down the above in case anyone else runs into this sort of thing.