Briefly, if the patient is a smoker, always think of:
- Chronic obstructive pulmonary disease (COPD)
- Lung cancer
On the other hand, if the patient isn't a smoker, think mainly of:
- Postnasal drip syndrome
- Gastroesophageal reflux
And for either category, remember to ask the patient whether or not they're using an angiotensin-converting enzyme (ACE) inhibitor (e.g. for hypertension, cardiac failure or diabetes). This is a also a common cause of a chronic cough.
Of these, the 'non-smoker' three, plus the ACE inhibitor one, are the commonest causes of a chronic cough with a normal chest X-ray. The 'smoker' two are more likely to give you an abnormal X-ray, but in their earliest stages, their changes might not be apparent radiologically.
As I said, I'm deliberately only concentrating on the commonest causes. If none of these seem to fit, I'd ask for expert (i.e. pulmonologist) assistance. But that's just me...