The question already has good answers but I think there is still more to be said.

As already mentioned, algebraic K-theory satisfies Zariski descent.  For regular noetherian schemes this is due to

* Kenneth S. Brown, Stephen M. Gersten, _Algebraic K-theory as generalized sheaf cohomology_, Higher K-Theories, Lecture Notes in Mathematics Volume 341, 1973, pp 266-292.

It was generalized to finite dimensional noetherian schemes by

* R. Thomason, _Higher algebraic K-theory of schemes and of derived categories_, The Grothendieck Festschrift, vol. III, Progress in Mathematics, vol. 88, Birkhäuser, Basel, 1990, pp. 247–435.

However, there is a stronger statement: it satisfies descent with respect to the Nisnevich topology (which lies between Zariski and etale).  This is due to

* Yevsey A. Nisnevich, The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory, Algebraic K-theory: connections with geometry and topology, 1989, pp 241-341.

and was generalized to finite dimensional noetherian schemes in the same paper of Thomason.

In the following paper the above results are extended to finite dimensional quasi-compact quasi-separated schemes.

* Andreas Rosenschon, P.A. Ostvær, _Descent for K-theories_, Journal of Pure and Applied Algebra 206, 2006, pp 141–152.

I learned from Peter Scholze that, in modern language, a proof can be given with  the following main ingredients: 1) an identification of $K(X)$ with the K-theory of the stable infinity-category $\mathbf{Perf}(X)$ of perfect complexes; 2) descent for the presheaf of infinity-categories $\mathbf{Perf}(-)$; 3) a characterization of extendibility of perfect complexes on open subschemes; and 4) a homotopy fibre sequence of connective spectra $K(X \text{ on } Z) \to K(X) \to K(U)$ where $Z \subset X$ is the closed complement of an open subscheme $U \subset X$.  By 3) I mean the following statement: a perfect complex on an open subscheme $U \subset X$ can be extended up to quasi-isomorphism to a perfect complex on $X$, if and only if its class in $K_0(U)$ lies in the image of $K_0(X)$.  I am not sure in exactly what generality this proof works.

Finally, let me mention that the question of etale descent is closely related to the Lichtenbaum-Quillen conjecture.  This is now a theorem of Rost and Voevodsky and it implies that K-theory does satisfy etale descent in sufficient large degrees.  The theorem of Trobaugh that Steven Landsburg mentioned, about etale descent for mod-$\ell$ Bousfield-localized K-theory, is in

* R. Thomason, _Algebraic K-theory and étale cohomology_, Ann. Sci. Ecole Norm. Sup. 18 (4), 1985, pp. 437–552.

and is also generalized in the paper of Rosenschon and Ostvær.