Fluid mechanics plays a vital role in early vertebrate embryo development, an example being the establishment of left–right asymmetry. Following the dorsal–ventral and anterior–posterior axes, the left–right axis is the last to be established; in several species it has been shown that an important process involved with this is the production of a left–right asymmetric flow driven by ‘whirling’ cilia. It has previously been established in experimental and mathematical models of the mouse ventral node that the combination of a consistent rotational direction and posterior tilt creates left–right asymmetric flow. The zebrafish organizing structure, Kupffer’s vesicle, has a more complex internal arrangement of cilia than the mouse ventral node; experimental studies show that the flow exhibits an anticlockwise rotational motion when viewing the embryo from the dorsal roof, looking in the ventral direction. Reports of the arrangement and configuration of cilia suggest two possible mechanisms for the generation of this flow from existing axis information: (a) posterior tilt combined with increased cilia density on the dorsal roof; and (b) dorsal tilt of ‘equatorial’ cilia. We develop a mathematical model of symmetry breaking cilia-driven flow in Kupffer’s vesicle using the regularized Stokeslet boundary element method. Computations of the flow produced by tilted whirling cilia in an enclosed domain suggest that a possible mechanism capable of producing the flow field with qualitative and quantitative features closest to those observed experimentally is a combination of posteriorly tilted roof and floor cilia, and dorsally tilted equatorial cilia.