A close friend of mine also struggles chronic achilles tendinitis. His condition was initially caused by a running injury years ago and now flares up when he's on the bike (go figure). One leg is 1/4 inch shorter than the other, which was probably at the root of the problem, and if he doesn't back off when the condition starts the pain can affect his knee and hip. Rest, hydration and diet are his solutions, along with orthotics. I agree with Knotted as well, when you're on the bike you move around a lot and probably unconsciously get into less stressful positions.
If the problem continues, I suggest seeing an experienced sports PT.