The electrical properties of and the magnetoresistive effect in RCu3Mn4O12 (R=rare-earth ion or Th) are studied. In all compounds of this series, the magnetoresistive effect amounts to 20% at liquid nitrogen temperature in the presence of a field of 0.9 T. An increase in the magnetoresistance with decreasing temperature and a high sensitivity to weak magnetic fields at low temperatures point to the intergranular nature of the effect. The magnetoresistance shows a peak in the vicinity of the Curie temperature TC. Based on the dependences of the magnetoresistance on an external magnetic field, it is assumed that the magnetoresistance peak near TC is related to the charge carrier scattering by magnetic inhomogeneities as in substituted orthomanganites. We believe that the magnetoresistance value near the magnetic ordering temperature depends on the synthesis conditions and the effect of the intergranular spacer on the transport properties of these compounds.