If the court is listed as Circuit, County, Criminal, or Superior then the record is held by the Cook County Circuit Court Archives. If the court is listed as District, then NARA Great Lakes has the record. But, what if no court is listed?
Recently I had a client send me the two cards below and I wasn't sure how to follow up because no courts were given. (I removed the names for posting.)
A quick email to NARA solved the mystery. The "730" prefix is the code for the United States District Court in Chicago.
If you visit NARA's page on naturalizations you'll find a link that will let you order copies of naturalization records online for $7.50 (includes postage).
If you need naturalization records from the Circuit Court Archives, you can submit a request by mail or I can retrieve them for you for a $10.00 fee.
Many of these records are also available on Family History Library films. Search the catalog to check for availability.