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In this talk we present our construction of a characteristic map, using a "SAYD-twisted" cyclic cocycle, by which we transfer the characteristic classes of transversely orientable foliations into the cyclic cohomology of a certain noncommutative algebra. We carry out the explicit computation in codimensions 1 and 2. In codimension 1, we show how our result matches with the (only explicit) computation done by Connes-Moscovici, and in codimension 2 we present the transverse fundamental class, the Godbillon-Vey class, and the other four residual classes as cyclic cocycles. This is a joint work with B. Rangipour.