Over the last few years, the pre-decreases or pre-increases of the cosmic-ray intensity observed before a Forbush decrease, called the precursor effect and registered by the worldwide neutron monitor network, have been investigated for different cases of intense events. The Forbush decreases presented in this particular study were chosen from a list of events that occurred in the time period 1967 -2006 and were characterized by an enhanced first harmonic of cosmic-ray anisotropy prior to the interplanetary disturbance arrival. The asymptotic longitudinal cosmic-ray distribution diagrams for the events under consideration were studied using the "Ring of Stations" method, and data on solar flares, solar-wind speed, geomagnetic indices, and interplanetary magnetic field were analyzed in detail. The results revealed that the use of this method allowed the selection of a large number of events with well-defined precursors, which could be separated into at least three categories, according to duration and longitudinal zone. Finally, this analysis showed that the first harmonic of cosmic-ray anisotropy could serve as an adequate tool in the search for precursors and could also be evidence for them.