  It is well-known for an elliptic curve with complex multiplication that the
existence of a $\mathbb{Q}$-rational model is equivalent to its field of moduli
being equal to $\mathbb{Q}$, or its endomorphism ring being the ring of
integers of 9 possible fields ($\ast$). Murabayashi and Umegaki proved
analogous results for abelian surfaces. For three dimensional CM abelian
varieties with rational fields of moduli, Chun narrowed down to a list of 37
possible CM fields. In this paper, we show that his list is exact. By
constructing certain Hecke characters that satisfy a theorem of Shimura, we
prove that precisely 28 isogeny classes of these abelian varieties have
$\mathbb{Q}$-models. Therefore the complete analogy to $(\ast)$ fails here.
