The magnetic ordering and critical behavior of antiferromagnetic Fe(NCS)₂(pyrazine)₂ has been studied by neutron powder diffraction (NPD), inelastic neutron scattering (INS), Mössbauer spectroscopy, and magnetic measurements. The system can be regarded as a two-dimensional (2D) antiferromagnet even in the ordered phase, given that long-range magnetic ordering between the layers simply follows a necessary consequence of the establishment of long-range ordering within the planes. The INS data, which were taken on a cold neutron time-of-flight spectrometer, reveals that when the temperature is lowered towards T_N, the correlation length within the 2D layers increases and ultimately crosses over from two- to three-dimensional (3D) behavior. Indeed, 3D long-range antiferromagnetic order, associated with a propagation vector [1,0,1/4+ε], is observed in the NPD data below 6.8 K. Furthermore, in agreement with the behavior of both χ(T) and C_m(T) data, the order parameter follows the exact Osanger solution for a 2D S=1/2, Ising system.