The key requirement for successively running clones is that host buffers are flushed so that all data resides on the source LUN (and thus on the clone LUN) before the fracture is initiated. You can achieve this using various tools on Windows but is more difficult to accomplish on other operating systems (with the possible exception of Solaris).
Unmounting the source LUN before a fracture is not required because it is not necessary if the host buffers can be flushed using other tools. Further, it may not be wanted by the customer. Application like Oracle, SQL, or Exchange provide for hot backups, which would be defeated by the requirement to unmount source LUN. Therefore, as long as the user has ensured that the host buffers are flushed to disk properly, there should not be any need to unmount the source LUNs.
Note: It is also recommend that I/O to the source be quiesced during a fracture. There have been cases where this was not done and as a result, FSCK errors were encountered. If the LUN was unmounted, these errors would stop as an unmount ensures that no I/O is going to the source. However, this would not be necessary if I/Os had been quiesced.