I have a list of chicken resistant plants that I have been working on. My chickens are generally out in the backyard from dawn to dusk. I don't keep my hens confined to a run very much, so my list might be completely useless if you are trying to keep plants a run.
https://www.backyardchickens.com/web/viewblog.php?id=20741-plants-chickens-dont-eat
I have a similar list on my blog with lots of garden photos that might have some more additions since I made this BYC one last year. In general, plants in the salvia (mint) family are left alone by my chickens. I use some of the methods mentioned by others on this thread as well, like fencing off tender plants until established, wire mess on the ground around root zones to keep them from digging out newly planted plants, permanent fencing on the vegetable garden.
I agree, that the load of chickens has a lot to do with the amount of impact they have on a yard. I have 3 chickens in a backyard space of about 2000 square feet. I had 6 last summer, but between too many chickens and the drought, the lawn suffered. As soon as I downsized, the lawn and yard bounced back. 3 LF hens seems to be just about right for the space. Other than a few holes for dust bathing, my yard is nice. Also, what plants my hens don't like, might be different from someonelse's. When I buy a new plant, I only get one, and then leave it in the chickens' yard for a few days. I wait to see if they like it or not before bothering to plant it and buying a few more. Deer resistant plant lists is what got me started on my list; I noticed there were lots of duplicates to what my hens were ignoring at my house or my neighbors'.
*Now, if there is nothing else green for your chickens to eat in their yard, they will mow just about anything down, and might even eat the poisonous plants they would otherwise avoided.