The total energy of the transverse spin-spiral wave as a function of the wave vector for all transition metal atomic chains has been calculated within ab initio density functional theory with the generalized gradient approximation. It is predicted that at the equilibrium bond length, the V, Mn, and Fe chains have a stable spin-spiral structure, while the magnetic ground state of the Cr, Co, and Ni chains remains collinear. Furthermore, all the exchange interaction parameters of the transition metal chains are evaluated by using the calculated energy dispersion relations of the spin-spiral waves. Interestingly, it is found that the magnetic couplings in the V, Mn, and Cr chains are frustrated (i.e., the second near-neighbor exchange interaction is antiferromagnetic), and this leads to the formation of the stable spin-spiral structure in these chains. The spin-wave stiffness constant of these metal chains is also evaluated and is found to be smaller than its counterpart in bulk systems. The upper limit (on the order of 100 Kelvins) of the possible magnetic phase transition temperature in these atomic chains is also estimated within the mean-field approximation. The electronic band structure of the spin-spiral structures have also been calculated. It is hoped that the interesting findings here of the stable spin-spiral structure and frustrated magnetic interaction in the transition metal chains will stimulate further theoretical and experimental research in this field.