Multichannel Efimov physics is investigated in ultracold heteronuclear admixtures of K and Rb atoms. We observe a shift in the scattering length where the first atom-dimer resonance appears in the K-41-Rb-87 system relative to the position of the previously observed atom-dimer resonance in the K-40-Rb-87 system. This shift is well explained by our calculations with a three-body model including van der Waals interactions, and, more importantly, multichannel spinor physics. With only minor differences in the atomic masses of the admixtures, the shift in the atom-dimer resonance positions can be cleanly ascribed to the isolated and overlapping Feshbach resonances in the K-40-Rb-87 and K-41-Rb-87 systems, respectively. Our study demonstrates the role of multichannel Feshbach physics in determining Efimov resonances in heteronuclear three-body systems.