I found a very simple guide for recovering data from an encrypted drive in the scenario where they drive will not boot (but, obviously, where the drive is still functional).

Take a look:

How to Recover Data from an Encrypted Harddisk on Boot Failure with Ubuntu 14.04

This method may work with other related versions of Ubuntu (assuming they are using the same application for encryption).  I have not tested it, but it looks sound enough.

Let me know your experiences.  Hopefully this will help someone not lose data.