In the presented paper, an approach for the direct triangulation of 3D surfaces has been applied to the family of discrete surfaces represented by a triangular grid of arbitrary topology. The limit surface is reconstructed using a subdivision technique. An interpolating subdivision scheme based on the modified Butterfly scheme has been adopted. This scheme yields the differentiable limit surface. The actual discretization is based on the advancing front technique constrained directly to the limit surface. Various aspects concerning the performance of the algorithm have been addressed. Although the employed interpolating scheme exhibits several advantages, including locality and simplicity, numerical experiments still reveal relatively high computational demands of this technique (compared to the performance of the similar algorithm working on parameterized surfaces) and significant sensitivity to poorly shaped elements in the control grid. The latter drawback also limits the applicability of this technique to surfaces described by the Stereo Lithography (STL) grids without special treatment that is the subject of further research.