The dynamics of straight-line flows on compact translation surfaces (surfaces formed by gluing Euclidean polygons edge-to- edge via translations) has been widely studied due to connections to polygonal billiards and Teichmüller theory. However, much less is known regarding straight-line flows on non-compact infinite translation surfaces. In this talk we will review work on straight line flows on infinite translation surfaces and consider such a flow on the Mucube – an infinite $\mathbb{Z}^3$ periodic half-translation square-tiled surface – first discovered by Coxeter and Petrie and more recently studied by Athreya-Lee. We will give a complete characterization of the periodic directions for the straight-line flow on the Mucube – in terms of a subgroup of $\mathrm{SL}_2 \mathbb{Z}$. We will use the latter characterization to obtain the group of derivatives of affine diffeomorphisms of the Mucube. This is joint work (in progress) with Andre P. Oliveira, Felipe A. Ramírez and Chandrika Sadanand.