To start with – as little CORN and WHEAT as possible.  Much much more necessary for cats than dogs, but as one of the articles I’ve read put it – “When was the last time you saw a cat or dog hunt a loaf of bread?”
Possible ingredient listings:  wheat bran, wheat flour, wheat germ meal, wheat mill run, wheat shorts, middlings, wheat red dog, defatted wheat germ meal, corn bran, corn feed meal, cracked corn, ground corn, corn grits, corn flour, hominy feed, maltodextrins…  etc…
(Other grains aren’t lovely either, so if you see:  barley, grain sorghum, oats, or any kind of screenings, aspirated grain fractions….)
Generally, try to make sure grain is not in the top section of the list of components for what you are feeding your pet.  Brewer’s rice is another ingredient to stay away from if possible.  Applies for humans, too.  This is the small broken grains left over after rice has been processed.  No nutritional value, just empty calories.  If curious for a lot more data, visit www.petsfortheenvironment.org.

This next bit is fairly disgusting.  Stay away from “meat meal.”  This means meat has been rendered and dried and than added into the food.  That part doesn’t sound too awful.  However, meat meal can contain both dead zoo animals AND the corpses of euthanized pets.  For me, the concept of feeding my cats and dogs dead cats and dogs is rather enough.  However, those corpses still contain amounts of the drugs used to euthanize the animals to begin with, which means the food will as well.  Even short term testing has indicated that this is probably not a good idea, but over a lifetime the effects likely snowball.  Other things that can be included in meat meal are restaurant grease and supermarket waste.

Due to my research over time into this matter, in a perfect world, I would make the food for them myself….  not having that kind of energy, we make sure that we know what all the ingredients are, what they mean, and why they are included.