How can I get rid of chest congestion that has persisted a month after having the flu?

This is an unfortunate problem that you are still dealing with. As you know, what is commonly called the flu can have nasty consequences, which is why the flu vaccine is so much recommended by the health care profession. Fortunately, you seem to be doing better. Hopefully, you have followed up with your primary care doctor to confirm that you are doing well and that there are no further medical concerns. If not, the fact that you are still having symptoms this far out does warrant a visit to him or her. With regards to chest congestion, the things that will help to clear out your lungs will be those exercises that require you to breathe deeply often. An incentive spirometer is used frequently in the hospital to expand the lungs, but adequate physical activity that gets you moving is likely even better (provided that it is safe for you to do so after talking with your physician). As for your vocal cords and the problems you seem to be having, the most likely cause of your complaints, if it has anything to do with the flu, relates to the continued mucous drip that is irritating your throat and the vocal cords. This can cause continued hoarseness and a cough for some time until it resolves completely. Please see your physician to confirm that there is nothing else occurring, and for further advice to help you feel better.