Grant read/read-write access to SSAS cubes
Writeback permissions are a bit tricky. I documented them here. The relevant part is:


  Next, build a separate Writeback role. On the Cubes tab, grant
  Read/Write, check Process, and (optionally) Drillthrough permissions.
  I’m frankly not sure why you need Process permissions to perform ROLAP
  writeback, but it appears you need Process permissions for ROLAP or
  MOLAP writeback.
  
  In the Data Sources tab, make sure to grant Read permissions on the
  data source used for writeback. For example, when I created my
  writeback partition, I used the Greg Test data source to direct that
  writeback table to live in another database (such as a database with
  Full recovery model).


Sounds like you already figured it out but we can document this for posterity.