Your grandmother's naturalization petition will probably have the most details. You can request it online at http://www.archives.gov/http://www.archives.gov/ - click the "order online" button in the left margin. If they do not have it on file they will not charge you for it. You may also be able to get it from the clerk of the county court where it was initially filed.
Her wedding license would also be a possible source but sometimes the bride and groom listed only the country.
Death certificates also request birthplaces but this information if usually wrong or missing because the child who is giving the information does not know.