Recently, quaternion orthogonal designs (QODs) were introduced as a mathematical construct with the potential for applications in wireless communications. The potential applications require new methods for constructing QODs, as most of the known methods of construction do not produce QODs with the exact properties required for implementation in wireless systems. This paper uses real amicable orthogonal designs and the Kronecker product to construct new families of QODs. The proposed Amicable-Kronecker Construction can be applied to build quaternion orthogonal designs of a variety of sizes and types. Although it has not yet been simulated whether the resulting designs are useful for applications, their properties look promising for the desired implementations. Furthermore, the construction itself is interesting because it uses a simple family of real amicable orthogonal designs and the Kronecker product as building blocks, opening the door for future construction algorithms using other families of amicable designs and other matrix products.