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We study the resolution of an Ulrich bundle of arbitrary rank on the Segrefourfold $\PP^2\times\PP^2$. We characterize the Ulrich bundles $\Vv$ ofarbitrary rank on $\PP^2\times\PP^2$ with$h^1(\Vv\otimes\Omega\boxtimes\Omega)=0$ or with$h^2(\Vv\otimes\Omega(-1)\boxtimes\Omega(-1))=0$ or obtained as pullback from$\PP^2$ and we construct more complicated examples.

Ulrich bundles on the degree six Segre fourfold