  Let $X$ be a projective K3 surface over $\mathbb C$. We prove that its Cox
ring $R(X)$ has a generating set whose degrees are either classes of smooth
rational curves, sums of at most three elements of the Hilbert basis of the nef
cone, or of the form $2(f+f')$, where $f,f'$ are classes of elliptic fibrations
with $f\cdot f'=2$. This result and techniques using Koszul's type exact
sequences allow to determine a generating set for the Cox ring of all Mori
dream K3 surfaces of Picard number three which is minimal in most cases. A
presentation for the Cox ring is given in some special cases with few
generators.
