Based on the rules of conventional croquet "Bicycle Croquet" probably came about in and around Graz/Austria in the beginning of the 20th century. The modern variation of Bicycle Croquet(from German "Fahrradkrocket") has been played since 1997, when Mike Fugeman (England) and Wolfgang Wendlinger (Austria) reinvented the Sport in Aigen im Ennstal/Austria. (too much alcohol at too high altitude, and a bunch of english students)
The homepage of BCCGraz (Bicycle Croquet Club Graz) gives the following description of the sport:
Bicycle Croquet - the game
In addition to a high level of fairness BC demands strict control over a player's body and soul. The correct control over ones bicycle is essential for an exciting and stimulating BC-game. The Rund (the ball) is played through the gates by the Holz (mallet). A time limit of 10 seconds has to be met for every individual play. The players alternate strokes and are not allowed to touch the ground with any part of their body. He or she who manages first to hit the peak at the end of the course with their runds wins the game.
Graz Club: http://www.bccgraz.org/