We have fabricated pillar-like microstructures of Fe---Cr and Co---Cu magnetic multilayers and measured the giant magnetoresistance effect with the current perpendicular to the multilayer plane. Lithographic and reactive ion etching techniques were used to define "pillars" with a height of typically 0.5 µm and a width ranging between 3 and 10 µm. The perpendicular current density pattern in these structures was modelled analytically, providing an accurate description of the scaling of the experimental resistance with pillar size. The perpendicular giant magnetoresistance effect was determined as a function of temperature from 4 to 300 K. For both Fe---Cr and Co---Cu multilayers, we found, at low temperature, magnetoresistance effects of the order of 100%. The Fe---Cr pillars showed a pronounced decrease in magnetoresistance with temperature, whereas for Co---Cu the temperature dependence was much weaker; this is due to the fact that, for our Co---Cu multilayers, the thermally induced scattering is more spin dependent than for Fe---Cr. Finally, we compared our microfabrication approach of the perpendicular magnetoresistance experiment with the superconducting contacting technique developed at Michigan State University.