Each country likely has its own procedures. But here's my experience.
Earlier this year, we tried to get Italian citizenship for my husband Tom. His grandfather Cornelius came to the US in 1903 and married in 1909. In 1910 he had a son (my husband's father) and in 1911 Cornelius became a naturalized American citizen.
Because Tom's father was born before Cornelius was naturalized, Tom would have been eligible for Italian citizenship ... but he would have had to apply within 100 years of his father's birth. So we missed it by 3 years!
In addition to an application, we would have needed to submit some paperwork (birth certificates for father and grandfather, naturalization papers for grandfather, etc.) so we may still have run into difficulties even had we been within the time period.
Before you do a lot of work, you might call the French consulate in Chicago, and hopefully you will find a helpful person who can quickly tell you if you even qualify. That's what I did; I was running into difficulties with the Italian birth certificate and called to ask a question, which is when I found out about the 100 year time limit. I did, of course, call back and verify that fact with another person, just in case.
At least for the Italians, the key information was when your grandparent was naturalized, and when your parent was born. If your grandparent became a naturalized American citizen before your parent was born, you would be out of luck.
I found Ancestry.com to be invaluable when doing my research, and ended up with a lot of really interesting family history, albeit no dual citizenship.
Good luck!