It has taken some time, but since several weeks OMS / Azure Log Analytics is integrated into Azure Resource Manager and with this comes the capability to move your workspace between different resource groups.
Open the Azure portal (https://portal.azure.com) and go to your OMS workspace.
Just click on the pen behind the resource group name and a new blade opens. Choose the new resource group and enter OI-De as confirmation. As you may have seen my OMS workspace has three connected storage account logs. They will play an important role during the move.
After an hour my move task failed without an informal error message. So I did a retry via PowerShell and again it failed, but now I got an error message with the necessary information.
…The workspace ‘Azure-Stack-OMS’ cannot be moved unless all storage configurations are also moved or deleted…
Currently and as far as I know you cannot move a storage account between different resource groups. So before my second retry I did a cleanup of the connected storage account logs and voilà it worked this time.