The aim of this paper is to study the characterization of some constant mean curvature (CMC) and constant Gauss curvature (CGC) A-net surfaces in the 3-dimensional Lorentz-Heisenberg group H3 endowed with a left invariant Lorentzian metric gi in the three following cases :  1. g1 = -dx2 + dy2 + (xdy + dz)2, 2. g2 = dx2 + dy2 - (xdy + dz)2. 3. g3 = dx2 + (xdy + dz)2 - [(1 - x)dy - dz]2. Using Levi-Civita connection, we will calculate the mean curvature H and the Gaussian curvature K in order to study CMC and CGC A-net surfaces in (H3, g1), (H3, g2)  and (H3, g3).