In theory, it would be possible to take the results of an MDX query and import that into a table. I don't think there are any built-in tasks in SSIS that extracts MDX into a result set, but you would be able to do it with a .NET script. Could be messy though...

I don't think it would be possible to extract the contents of an entire cube into relational tables. Remember that a cube also stores calculations and aggregated values, which just would not fit into the two-dimensional structure of a relational database.

What you could probably do as well, is look at the data source view of the SSAS project as well as the code for the calculated members...and recreate that in Cognos. Not sure if it would be worth the amount of effort though...