we prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3-dimensional riemannian manifolds. the constructed min-max surfaces are smooth with at most one singular point. the constant anisotropic mean curvature can be fixed to be any real number. in particular, we partially solve a conjecture of allard [invent. math.,1983] in dimension 3.