What about ww1 bicycle troops? WW1 was one of the only wars that really put them into effect, and certain rulesets actually suggest that you include them in army lists.
British: http://freepages.genealogy.rootsweb.ancestry.com/~westaust/Postcards/Military/Graphics/X%20WW1%20BICYCLE%20RIFLE%20CORP-1.JPG
German: http://1.bp.blogspot.com/-W8CBXu0Sx-w/Th5e8KGY0II/AAAAAAAAADw/dY40UY_ESok/s400/normal_02519.jpg
"one of the only...." - always found that one of the strangest Americanisms.
The only cyclist I can find is GR45 in the WW2 German. But it is possible as you can see.
http://www.pendraken.co.uk/GR45-p4324/#img (http://www.pendraken.co.uk/GR45-p4324/#img)
Oh, and welcome to the forum, Nousforausername.
2/KSLI came ashore on DDay with shinny red bicycles as part of the Second Intermediate Assault wave
Then promptly dumped them on the beach! The Battalion commander and W Company CO kept there's as they were observed riding them between hot spots later in the afternoon.
The French locals were riding the bikes for years after!
Hello Nousforausername
The Japanese also have a cyclist:-
(http://farm5.static.flickr.com/4101/4750696846_91483af2f0.jpg)
Cheers and welcome
GrumpyOldMan
WWI cyclists get my vote - particularly German Jaegers
German cycle troops (ww1):
http://en.wikipedia.org/wiki/File:HJB10_%E2%80%93_Radfahr-Kompanie.jpg
Italian Cycle troops (ww1):
http://en.wikipedia.org/wiki/File:BicycleBersaglieri2.jpg