  We establish the group-theoretic classification of Sato-Tate groups of
self-dual motives of weight 3 with rational coefficients and Hodge numbers
h^{3,0} = h^{2,1} = h^{1,2} = h^{0,3} = 1. We then describe families of motives
that realize some of these Sato-Tate groups, and provide numerical evidence
supporting equidistribution. One of these families arises in the middle
cohomology of certain Calabi-Yau threefolds appearing in the Dwork quintic
pencil; for motives in this family, our evidence suggests that the Sato-Tate
group is always equal to the full unitary symplectic group USp(4).
