Do you spread mulch over your garden? If not, that should definitely help. A good 3- to 4-inch-deep layer of mulch should keep the vast majority of forget-me-nots from coming back.
You can also kill just the forget-me-nots by taking a paintbrush, dipping it in an herbicide such as Roundup, and then ""painting"" each forget-me-not. It's less work than pulling them and should kill them.
I hope this helps. Good luck!
---Justin W. Hancock, Senior Garden Editor, BHG.com, Better Homes and Gardens