Cyclic nucleotide-gated (CNG) ion channels are nonselective cation channels, essential for visual and olfactory sensory transduction. Although the channels include voltage-sensor domains (VSDs), their conductance is thought to be independent of the membrane potential, and their gating regulated by cytosolic cyclic nucleotide-binding domains. Mutations in these channels result in severe, degenerative retinal diseases, which remain untreatable. The lack of structural information on CNG channels has prevented mechanistic understanding of disease-causing mutations, precluded structure-based drug design, and hampered in silico investigation of the gating mechanism. To address this, we built a 3D model of the cone tetrameric CNG channel, based on homology to two distinct templates with known structures: the transmembrane (TM) domain of a bacterial channel, and the cyclic nucleotide-binding domain of the mouse HCN2 channel. Since the TM-domain template had low sequence-similarity to the TM domains of the CNG channels, and to reconcile conflicts between the two templates, we developed a novel, hybrid approach, combining homology modeling with evolutionary coupling constraints. Next, we used elastic network analysis of the model structure to investigate global motions of the channel and to elucidate its gating mechanism. We found the following: (i) In the main mode of motion, the TM and cytosolic domains counter-rotated around the membrane normal. We related this motion to gating, a proposition that is supported by previous experimental data, and by comparison to the known gating mechanism of the bacterial KirBac channel. (ii) The VSDs could facilitate gating (supplementing the pore gate), explaining their presence in such 'voltage-insensitive' channels. (iii) Our elastic network model analysis of the CNGA3 channel supports a modular model of allosteric gating, according to which protein domains are quasi-independent: they can move independently, but are coupled to each other allosterically.