The syntheses, structures, and characterization of six Ln(3+)-Cu(2+)-glycine (Hgly) coordination polymers are described in this paper. They represent three types of structures. Type I (Ln=La (1), Pr (2), and Sm (3)) is a 1D catenarian polymer comprising [Ln(2)] nodes bridged by four cis-Cu(gly)(2) linkers. Type II (Ln=Eu (4) and Dy (5)) is a 2D open framework with a 4(4)-net, composed of novel [Ln(6)Cu(22)] cluster nodes linked by trans-Cu(gly)(2) linkers. Furthermore, the inner structures of the [Ln(6)Cu(22)] nodes, and the connection mode between the nodes and linkers are slightly different for 4 and 5. Type III (Ln=Er (6)) is a 3D open framework with a novel 3(6)4(18)5(3)6 topology, made up of [Er(6)Cu(24)] cluster nodes and trans-Cu(gly)(2) linkers. The rich variety of the resulting structures owes itself mainly to the interselection between the dynamic control of metalloligands and cationic components. A transition from frequency dependence to frequency independence is observed in the field-induced magnetization lag for 1-3. The frequency dependence at low temperatures may come from the antiferromagnetic Cu--Cu interaction through the [Ln(2)] nodes, whereas the frequency independence may be due to the disappearance of the antiferromagnetic Cu--Cu interaction at high temperatures.