I can think of several ways to do this.

The first is to reflect the design into SSAS, and let the cube handle the security. While I'm only an SSAS novice, this is one of the things that it does well from memory. Users should only see what they are permitted to see.

The second is to create schemas for the users and in each schema, create views that filter out the data they are not permitted to see. Grant select rights on the views, not on the underlying tables.

The third is a variation on the second, and I think is better in terms of management. That is to create "subject areas" (again I would recommend schemas), create the views within the subject areas, assign the necessary permissions to the subject areas, and finally assign the users to the subject areas.