Why are there no dihedral mirror planes in the D3h point group?
The σ$_v$ planes in a molecule are a direct result of the C$_n$ main axis i.e. they are interchangeable if you perform the C$_n$ operation on them. On the other hand σ$_d$, which are called dihedral planes bisect the dihedral angles between members of the σ$_v$'s set. Therefore molecules with odd C$_n$ such as BH$_3$ wont have any σ$_d$ planes; only n (3 for BH$_3$) σ$_v$'s. In contrast a molecule such as benzene which has a C$_6$ main axis will have 3 σ$_v$'s (the ones including the carbon atoms) and 3 σ$_d$'s which are the ones passing between the carbon atoms. Of course it doesnt really matter which set is called dihedral and which is called vertical, it could be the other way round as well. Ref: Chemical applications of group theory by Cotton.