This paper studies the problem of the motion of a classical model of modern analytical multibody dynamics — a chain of gyrostats coupled by ideal spherical joints. The chain moves about a fixed point in a central Newtonian force field. The paper develops the equations of chain’s motion and then establishes and analyzes the conditions for existence of some classes of precessional motions of the chain, under the assumption that the barycenter of each gyrostat is located on the line connecting the points where it is attached to other gyrostats. The findings of the paper extend corresponding results in the dynamics of a single gyrostat to a case of the multibody chain as well as generalize some of the known properties of precessional motions in the dynamics of many bodies.