jags,
This may be a long shot but has your doctor checked you for a hiatal hernia?
The bent-over position while cycling combined with the exertion can worsen this condition if one has it, causing a reflux of stomach acid into the esophagus. If it is a persistent problem, I really would urge a doctor visit to check on the possibility of GERD (gastroesphageal reflux disease).
Some medications can make heartburn worse. My dad took some cholesterol-lowering meds by prescription and had to stop for that very reason. Diet can be a factor, as can weight for some people.
If you only get the feeling of heartburn during heavy exertion, then it really would be a good idea to see a doctor to rule out anything heart-related with a stress test or whatever.
Really, please take this seriously, jags.
All the best,
Dan. "not the MD kind of doctor, the other kind"