Sweden had bicycle-mounted infantry into the 1980:s (towards the end, mostly local defence units). However, the soldiers were not supposed to pedal for any significant distance (and certainly not to fight mounted!). They would be towed by tractors, and leave their steel steeds to fight on foot. I think the concept was introduced during the first half of the 20th century because it allowed increased mobility at a much lower cost than having everyone in a motor vehicle. It has also been noted that, in case of suddenly appearing aerial threats, a bicycle unit could melt away into the forest with remarkable effectiveness.