The copper(I)‐catalyzed alkine–azide cycloaddition (CuAAC) click polymerization is used for the synthesis of novel poly(dimethylsiloxane) (PDMS)‐containing alternating copolymers in a straightforward AA/BB polyaddition approach. The versatility of the CuAAC reaction allows incorporation of a series of different monomers varying in geometry, size, and functionality. A structure–property relation is revealed that shows that the polymer properties are dependent not only on the type of functionality incorporated but also on the size and geometry of the comonomer. The spectrum of properties that can be covered is highly interesting, ranging from viscous oils to brittle solids. Targeted application of monomers even gives access to properties that are typical for thermoplastic elastomers. Additional studies concerning the acceleration of the polymerization reaction support mechanistic considerations about a binuclear active species.