Four 1-cm(3) CdZnTe semiconductor detectors were operated in the Gran Sasso National Laboratory to explore the feasibility of such devices for double beta-decay searches as proposed for the COBRA experiment. The research involved background studies accompanied by measurements of energy resolution performed at the surface. Energy resolutions sufficient to reduce the contribution of two-neutrino double beta-decay events to a negligible level for a large-scale experiment have already been achieved and further improvements are expected. Using activity measurements of contaminants in all construction materials a background model was developed with the help of Monte Carlo simulations and major background sources were identified. A total exposure of 4.34 kg center dot days of underground data have been accumulated allowing a search for neutrinoless double beta-decay modes of seven isotopes found in CdZnTe. Half-life limits (90% C.L.) are presented for decays to ground and excited states. Four improved lower limits have been obtained, including zero neutrino double electron capture transitions of Zn-64 and Te-120 to the ground state, which are 1.19x10(17) years and 2.68x10(15) years, respectively.