Back pain while cycling is usually caused by poor cycling posture. Good cycling posture is very different from good posture while sitting or standing. A posture that is comfortable for sitting still will not necessarily be comfortable while actually riding a bicycle. Correct cycling posture must facilitate the pedaling action, and also must enable the rider to cope with the jolts that result from road irregularities.
When riding a bicycle, the back should be arched, like a bridge, not drooping forward between the hips and the shoulders. If the back is properly arched, bumps will cause it to flex slightly in the direction of a bit more arch; this is harmless. If you ride swaybacked, bumps will cause the back to bow even farther in the forward direction, which can lead to severe lumbar pain.
Some back-pain sufferers modify their bicycles with extra-high handlebars so that they can sit bolt upright, with their spines straight. This is actually counterproductive in most cases, because a straight spine has no way to "give" when the bike hits bumps. Road irregularities will jam the vertebrae together, often aggravating existing back problems. The bolt-upright posture is comfortable if you're sitting stationary on the bike, but is not suitable for riding much faster than a brisk walk. Riders who for some reason require such a position should use some form of suspension...a sprung saddle at the very least.