We study isoperimetric constants of and self-avoiding walks (SAWs) and self-avoiding polygons (SAPs) on Cayley graphs of hyperbolic Coxeter groups. These graphs are tilings of the hyperbolic plane. We calculate the isoperimetric constants of most rank three hyperbolic Coxeter groups and give an estimate for the remainder. We prove that for all but a few classes of these groups there are exponentially fewer k-step SAPs than there are k-step SAWs. We prove that this is also true of any nontrivial free product. Additionally we show that there are exponentially more k-step SAWs on a free group F_n (resp. a ?free? Coxeter group T_d) than on any strict subgroup of F_n (resp. T_d).