We prove a formula for the structure sheaf of a quiver variety in
the Grothendieck ring of its embedding variety. This formula
generalizes and gives new expressions for Grothendieck
polynomials. Our formula is stated in terms of coefficients that are
uniquely determined by the geometry and can be computed by an explicit
combinatorial algorithm. We conjecture that these coefficients have
signs that alternate with degree. The proof of our formula involves
K-theoretic generalizations of several useful cohomological tools,
including the Thom-Porteous formula, the Jacobi-Trudi formula, and a
Gysin formula of P. Pragacz.