  This article is about motives of quadric bundles. In the case of odd
dimensional fibers and where the basis is of dimension two we give an explicit
relative and absolute Chow-K\"unneth decomposition. This shows that the motive
of the quadric bundle is isomorphic to the direct sum of the motive of the base
and the Prym motive of a double cover of the discriminant. In particular this
is a refinement with $\mathbb Q$ coefficients of a result of Beauville
concerning the cohomology and the Chow groups of an odd dimensional quadric
bundle over $\mathbb P^2$. This Chow-K\"unneth decomposition satisfies Murre's
conjectures II and III. This article is a generalization of an article of Nagel
and Saito on conic bundles \cite{NS}.
