in spain "montar" is used, and a lot of south american countries prefer "andar", but there is still a lot of variation, either would be understood. just make sure you get your preposition right (to ride a horse uses the same verbs, but uses 'a' instead of 'en').

andar en bici/ montar en bici

edit: useful discussion in another forum... http://forum.wordreference.com/showthread.php?t=290227

i am also a language nerd.