Here's what a Mr. Laubin has to say on the subject:
"The English tradition is to wear the quiver over the right shoulder, and this is the tradition that modern archers carry on to this day. To me it is a clumsy arrangement, especially with the long arrows most people prefer. Plains quivers, carried over the left shoulder, could be quickly pulled around to the front so that the arrows either projected under the left arm, or they were pulled around still farther so that the quiver sling hung on the back of the neck and the arrows were at the waist with their nocks on the right-hand side, where they could be quickly reached while riding. Either way is far more convenient than the English method." (Laubin, Reginald. American Indian Archery. Norman: University of Oklahoma Press, 1980. p. 127)