The paper deals with linear one-dimensional integro-differential vibrating systems of the Timoshenko type with past history acting only in one equation. The authors show that the dissipation given by the history term is strong enough to produce the exponential stability of the considered systems if and only if the equations have the same wave speeds. Otherwise the corresponding system does not decay exponentially as time goes to infinity. In the case when the wave speeds of the equations are different, it is proved that the solution decays polynomially to zero, with rates that depend on the regularity of the initial data.