We have measured the giant magnetoimpedance (GMI) of microstructured Co-based Metglas (R) 2714a amorphous magnetic ribbons. Before micropatterning, the amorphous ribbons were chemically thinned or polished, to study the influence of sample thickness and surface state on the GMI effect. The magnetic domain structure of polished foils near saturation is observed using the bitter fluid technique. The well-defined geometry of the samples allows a quantitative comparison of the GMI effect with a simple skin depth model, from which we obtain the frequency-dependent permeability of the material. We have measured the GMI effect for magnetic fields parallel and transverse to the long axis of the sample. We compare the different measurement geometries and film preparation methods and, typically, find a room-temperature GMI ratio of about 35%-90% in fields of a few mT and at frequencies of 10-40 MHz.