Answer by Igor Rivin for Which surfaces admit unbounded-length simple geodesics?

This paper

Rouyer, Joël(R-AOS); Vîlcu, Costin(R-AOS)Simple closed geodesics on most Alexandrov surfaces. (English summary) Adv. Math. 278 (2015), 103–120. 53C45 (53C22) 

Indicates that this is usually true whenever curvature is not everywhere non-negative. In genus $1,$ it is ALWAYS true when the curvature is everywhere $0.$