Georgia registration of deaths statewide began in 1919 and was generally complied with by 1922. But there may be records at the local level. Unfortunately, marriage records in Georgia didn't record the parents of the bride and groom, so you won't be able to find direct evidence of his father's name there. You may have to build a case using indirect evidence, using techniques like the ones demonstrated in Dr. Thomas W. Jones' class Inferential Genealogy.
Dr. Jones explains that when we blindly accept the information given in a primary record, we are simply accepting the information stated in the record. But as he shows in his article Perils of Source Snobbery, primary sources can be wrong. We need to determine for ourselves if the stated relationships in a record are correct, or to answer the question in other ways if no single source contains the information we seek.
If you already know the cemetery, that's an excellent place to start. Is the cemetery a commercial one or a church cemetery? If it is associated with a church, that gives you a pointer to possible church records. If it is a public cemetery, see if you can find out who is listed as the plot owner and who else might be buried in the family plot, since those are usually other relatives. Do you know who is buried nearby?
If you haven't done so already, construct a timeline of your 3rd great-grandfather's life, putting all the events you know about in chronological order with a brief note about where you found the information. (You say he was pronounced dead by a doctor -- how do you know? If it's a story which has been passed down through the family, write a note as if you were explaining it to someone else, saying what you know and who told you, just to clarify everything in your mind.)
Keeping a list of sources with a Source Checklist helps you to keep track of what you've gathered already. Once you've assembled everything you already have, review it all again.
Start with a Map
When working in Georgia (or in any locality) it's important to be aware of the dates that county boundaries changed. The Newberry Library has a website for the Atlas of Historical County Boundaries. You can use the interactive map portion, and you have Google Earth, you can download the shape files. You can also find Georgia county chronologies. Wikipedia's articles on the modern counties will often tell you what counties are adjacent, and they'll have maps.
Study family groups, friends, associates, neighbors
Did your 3rd great-grandfather have siblings? Sometimes you can fill in the gaps in your information about a person by studying the entire family as a group.
See QuickLesson 11: Identity Problems & the FAN Principle for an example of what you can do by looking at a group instead of concentrating on just one named individual.
Another strategy you could use is a record checklist like the United States Record Selection Table in the FamilySearch Research Wiki. For records with information about parents and other family members, they suggest starting with:
Vital Records, Church Records, Census, Directories, Probate Records, Obituaries, Genealogies
and then to try
Bible Records, Newspapers, Emigration and Immigration, Adoptions
In the wiki, each one of these selections is linked to another article which will give you information and search tips for that kind of record.
It's a good idea to start with someone's death, where you have the most complete portrait of their life, and work backwards in time. But whatever line of inquiry you follow, try to move outwards from what you know rather than making large leaps. Be on the lookout for details that will make your 3rd-great-grandfather stand out from other people of the same name, especially if his name is a common one. This is where the FAN principle really shines -- the associates and neighbors, as well as family, will help distinguish him from his same-name sound-alikes.
Don't assume that people didn't get into the newspapers simply because they were poor and couldn't pay for a notice. In historical newspapers, editors were often on the lookout to fill 'the news hole' (whatever space was left over after the ads were placed) and news about our ancestors could appear in any town where relatives might be living, not just the immediate area where the deceased lived.
If you have access to newspapers from the period, look over entire issues of the papers to see how the town news is printed in the paper, and which towns are covered by the paper. Also check the newspaper directory at the Library of Congress to see which papers were published around that time, and check your online provider to see what papers are included in their collection. All it takes is a couple of missing issues for someone's obituary or funeral notice to be 'missing'.
For online collections, check the directory at The Ancestor Hunt, Georgia Online Historical Newspapers, or Wikipedia's List of online newspaper archives.