Under consideration in this paper is rogue waves on the general periodic
travelling waves background of an integrable extended modified
Korteweg-de Vries equation. The general periodic travelling wave
solutions are presented in terms of the sub-equation method. By means of
the Darboux transformation and the nonlinearization of the Lax pair, we
present the first-, second- and third-order rogue waves on the general
periodic travelling waves background. Furthermore, the dynamic behaviors
of rogue periodic waves are elucidated from the viewpoint of
three-dimensional structures.