I know exactly what you're on about, I'm the same.
Turns out I have a heart condition called Wolfe Parkinson White, or WPW, which is far more common than it seems. It's basically, and I quote my doctor, a supraventricular tachycardia (sp???). Instead of there only being one electrical pulse making the heart pump, there's a secondary smaller one. It's all fairly newly discovered thanks to the development of more sensitive ECG machines. I basically find that when I am calm, or relaxing, and now more often when I'm just doing day to day things, I get a really powerful heartbeat followed by a number of palpitations, that last about half an hour to a few hours.
I'm not saying this is what you have, but the symptoms are very similar. I don't get so much of the breathlessness unless I'm having actual palpitations, but I do find it harder to get back to my usual pulse rate after exercise.