You are correct- Azure Analysis Services only supports tabular models.
The best option for migrating MDX cubes at this time is going to be to use an Azure VM running SQL Server. There are several prebuilt images to start from, which ensures that the VM is configured properly to run SQL Server and allows for flexibility in your licensing model. If you want to build your own VM image, then adding the SQL Server IaaS Agent extension will unlock those tools- there is no cost for running the extension itself.
If your existing licenses include a hybrid benefit, you can make use of those licenses within Azure. If you don't have an existing license you would like to use, then you also have the option of using a pay-as-you-go model that bundles the cost of the license along with the VM in a per-second pricing model.