Is the thumping in my right ear something to be worried about. it goes with my heart beat and has been on/off for 6 months.?

Thumping sound -ear: This may be a type of tinnitis and or it may be related to your blood pressure or your vessels around the inner ear. You need to see your doctor for a check up and then probably be referred to a ENT specialist.

Should get checked: What you are describing is called pulsatile tinnitus. Usually it is due to a small blood vessel near the hearing nerve. Rarely it can be due to a benign growth or vascular lesion. Although the majority of my patients with pulsatile tinnitus end up having negative work ups you should be seen by an ENT doctor.