Even without obvious symptoms of post-nasal drip, you may want to consider trying out a combination antihistamine-decongestant medicine to see if it relieves your cough. Post-nasal drip sometimes causes very subtle symptoms in the nose and throat, and a cough can be the only obvious symptom of allergic irritation in the nose and throat.
Another very common cause of persistent cough is asthma. Usually asthma causes a fairly obvious wheezing noise with your breathing (this can sound like a harmonica note or a raspy noise), and it causes you to feel short of breath.
Do you sometimes make a wheezing noise or feel short of breath?