How would you implement OLAP cubes if the raw data is inconsistent?

Question

I have been reading up about OLAP cubes, and I've spoken to our data architect about OLAP implementation within our warehouses. However, i learned that in order for OLAP to be implemented, the data that we receive has to be consistent. However, the data that we've always been receiving aren't. For instance, we received various forms of data in Flat File, CSV, AS2, and 852s. Additionally, there's a lot of custom metrics and cleansing we do internally. What about be an alternate way around OLAP?