  In this paper we discuss four methods of proving modularity of Calabi--Yau
threefolds with $h^{12}=1$: existence of elliptic ruled surfaces inside
(Hulek-Verrill), correspondence with a product of an elliptic curve and a K3
surface (Livn\'e-Yui), correspondence with a (modular) rigid Calabi-Yau
threefold, and existence of an involution splitting the fourdimensional
representation into twodimensional subrepresentations.
  We apply these methods to prove modularity of 17 out of 18 double octic
Calabi-Yau threefolds for which "numerical evidence of modularity" was found in
the second Author's thesis.
  We observe that modularity holds for those elements in a pencil having some
additional geometric properties. In the proofs we use representations of the
considered Calabi-Yau threefolds as a Kummer fibration associated to a fiber
product of rational elliptic fibrations.
