I had horrible, horrible heartburn from the day I conceived almost till the day I gave birth. I resisted the Gaviscon option (foolishly) for months as I read it was not good to take. This turned out to be rubbish - my GP telling me that she had a bottle under her desk the whole time she was pregnant cured me of this misconception. However, before I gave in to the delights of litre bottles of peppermint flavour (aniseed is VILE), the only natural thing I found that cured it for a while was fresh pineapple. They are hard to come by and expensive but it certainly did the trick with me. Apparently tinned or pineapple juice doesn't work - has to be fresh. Also avoiding fatty food, eating well before you go to bed and 'sleeping' upright (ha ha) supported by lots of pillows helps.
Good luck all you sufferers out there - I am now 6 weeks pg with No2 and so far - touch wood - no sign of that dreaded bubbling acid in the throat feeling.....