The seismic design of a two-storey precast reinforced-concrete building structure equipped with viscous dampers is presented in this paper with twofold purpose. The first goal is to verify the applicability of a practical procedure for the identification of the mechanical characteristics of the viscous dampers which allow to achieve target performance levels, originally proposed by the authors for moment-resisting building frames, also with reference to "pendular" structures. The second goal is to investigate the effectiveness of the use of viscous dampers (as compared with traditional lateral-resisting stiff braces) for the seismic design of precast not moment-resisting concrete structures.