I read some posts where people said they had trouble finding a decent deli. If you ever get out to Wood Dale on Irving Pk past O'Hare try Helen's. It's an excellent Polish deli that's been there for years.
I haven't tried everything there but what I've had would definitely make me a regular if I lived closer. Excellent salami, fresh Polish rye bread, and vegetable salad with peas and diced potatos. They also have Westfalian ham which I believe they get from Köneman's in Volo (a very good German sausage maker/deli in the hinterlands).
Ability to speak Polish helpful, but not needed.
Helen's Deli 272 W. Irving Park Rd., Wood Dale
Helen's also has a restaurant farther east on Irving. I ate there years ago and the home cooked Polish food was very good, but not outstanding.