The problem is essentially the Sinai billiard. That takes place on a finite square table with a circular hole removed (=peg added). There are standard bounces off the straight edges as well as off the peg.
There is a standard procedure of "unfolding" across flat edges: you just take a reflected copy of the table across any flat edge. There is a correspondence between trajectories in the unfolded table and the original table (a reflection in the original table across a flat edge just becomes a straight trajectory in the unfolded table).
Repeating this, you obtain exactly the model in the question. Sinai gave a statistical analysis of the properties of the trajectory which must imply that the set of angles for which the trajectory remains bounded has measure 0.
