So how do you tell what dimension is associated with a measure?

Open up BIDS (SQL Server Business Intelligence Development Studio)
From the file menu select: Open > Analysis Service Database
and choose your databsae and server etc.
Let the old girl sort itself out and then in the Solution Explorer select your cube of interest.

You’ll get something like this interlinked view:

Just switch to the Dimension Usage tab:

and you can see that Department is NOT linked to most measures.