We examine the detailed relationship between recurrent cosmic ray depressions and corotating high-speed streams in the inner heliosphere near the ecliptic plane using counting rates from anticoincidence guards of instruments on the IMP 8, Helios 1, and Helios 2 spacecraft. These rates indicate the density of >60 MeV/amu ions with reasonable time resolution (~15 min) and high counting statistics. Essentially all corotating streams are accompanied by significant particle depressions. The particle decrease commences most frequently (~63% of events) at the leading edge of the stream which is typically colocated with the stream interface inside the corotating interaction region (CIR). If the depression starts ahead of the interface, there is usually an additional abrupt decrease at the interface. In ~61% of events the onset of the depression is closely associated with the onset of enhanced field turbulence which typically occurs near the stream leading edge. Minimum particle densities are generally found in the vicinity of the maximum solar wind speed in the high-speed stream. The density recovers during the declining phase of the stream. The observations are most consistent with modulation of the cosmic ray density in high-speed streams by the increase in solar wind speed. Enhanced scattering by turbulence in the CIR may also contribute near the onset of the depression, in particular in cases where the decrease commences ahead of the stream interface. The absence of a consistent relationship between the depressions and magnetic field enhancements suggests that localized particle drifts in the enhanced magnetic fields of CIRs do not produce the particle depressions. The observation of a depression is not dependent on the presence of a heliospheric current sheet crossing in the CIR, in contrast to previous reports.