A three-dimensional manifestly Poincaré-invariant approach to the relativistic three-body problem is developed that satisfies the requirement of cluster separability and at the same time does not lead to so-called spurious states devoid of physical meaning. It is shown that these requirements make it possible to fix the form of the operators of the two-body interactions. The problem is solved with allowance for the dependence of the interaction operators on the spectral parameter. This dependence is a manifestation of the structure of the particles in the three-body system (i.e., it reflects the circumstance that the complete Hilbert space of state vectors of the system includes not only three-body configurations of the original particles) and leads to the appearance of certain factors in the cross sections of physical processes. Two alternative formulations of the method are investigated. In the first formulation, equations are written down for the amplitudes of transitions between free-particle states. In the second formulation, the states of interacting particles in the two-body scattering channels are used as complete orthogonal bases. Partial-wave expansions of the equations with respect to states with given total angular momentum of the system in the helicity basis are made.