Given three regular space curves r1(t), r2(t), r3(t) for t in [0,1] that define a curvilinear triangle, we consider the problem of constructing a triangular surface patch R(u1,u2,u3) bounded by these three curves, such that they are geodesics of the constructed surface. Results from a prior study (Farouki et al., 2009a) concerned with tensor-product patches are adapted to identify constraints on the given curves for the existence of such geodesic-bounded triangular surface patches. For curves satisfying these conditions, the patch is constructed by means of a cubically-blended triangular Coons interpolation scheme. A formulation of thin-plate spline energy in terms of barycentric coordinates with respect to a general domain triangle is also derived, and used to optimize the smoothness of the geodesic-bounded triangular surface patches.