Do you cut the peppers open? I would expect the chickens to like the seeds better than the flesh, so cutting the peppers open will let them get to the seeds.
About the flesh of the peppers--I see several people already suggested cutting it into small pieces or using a blender to make it easy to eat.
Also, do be realistic about how much of what you are expecting your chickens to eat. Each chicken is only able to eat a certain amount of food in a day. And nutritionally, a bell pepper is not the same as a complete chicken feed. So even if your chickens do decide they like bell peppers, they may not eat as many peppers are you are able to bring home.
Root vegetables would have a similar texture problem: they are just hard for chickens to eat. You could consider cooking the root vegetables until they are soft. Yes, that's a lot of bother, so just leaving them to compost in the chicken run might make more sense. And if the composting things attract bugs and breed worms, then the chickens can eat the bugs and worms, so at least you get some benefit from them that way.