We prove a new formulation of the K-theoretic Pieri rule regarding multiplication of stable Grothendieck polynomials using iterated residues. We also deploy our method to establish straightening laws to transform Grothendieck polynomials corresponding to general integer sequences to linear combinations of those corresponding to partitions. The technique of iterated residues appears at once similar to raising operators; however, the connection to path integrals in the complex plane provides a different perspective.