We've been sewing on twill shirts for a long time, but lately we're getting more complaints about excessive puckering, especially on some crappy poplin shirts we have in our line. We've approached the issue on every level, from digitizing, backing, machine settings, and so on, but haven't come up with a very good solution. Any suggestions?

Ussually twill fabrics (used for shirts,overalls,working and protective clothing)does not make any problems for embroidering,au contrary...it is fabric that people loves to embroidering to.

Here you can have couple of issues...but i'm sure that you have checked them as expirienced embroiderer...but anyway let's check again:
1. Digitizing
-if is same design that you used to work before and it was okay ...than obviusly is nothing wrong with digitized design
-if is new design,try to reduce number of underlay stitches,or maybe even to take them away...that's dependable on type and size of design.Also check that you don't use very long Satin stitches,and also try to avoid unneccessary stitching such as stitching one lay over another(make "holes"in design)
2. Backing
- if your standard backing is not "holding"fabric strong enough,try to use some of self adhesive backing or even better if you have heat adhesive backing(use iron to stick backing to fabric)
-also try to put water-soluble foil on top of garment
3. Embroidery machine
- first thing...change the needles-maybe needles are damaged or maybe needles are to thick
-release thread tension
-check timing of your Take-up lever cam,maybe is out of proper timing so is tightening thread at wrong time,so that can result with poor(too loose/too tight) thread tension or puckering
-check your boobin case...maybe is your bobbin case damaged and bobbin is not turning smoothly...if is stucking,also can make thread tension to tight which can result as puckering

This is some things that i can remember of...hope that will help.

Good luck!