The best way I can think of doing this currently is to create Bookmarks. Then, once a new workbook wants to be created, open the bookmark, and edit the data source so that it points at a different data source.
This would be a great one to put in the Ideas section!
Thanks Tracy. I know this is a while later, but I've just spun around to trying this again. I did the hierarchies manually the first time, and now I'm switching data sources. (aside: The original workbooks were made against a local MySQL server that was loading dumps from the production DB2 server. Now I am trying to make a copy of those workbooks that run against the live server, instead of the dumps)
The upside to the bookmark method is that the report that is bookmarked, then has the data source replaced, does still stay correct. The downside to the bookmark method is that it loses the hierarchies. In other words, I have a "project type" hierarchy in the original data source with three elements in it. I can use those elements in a report, then bookmark it. When I load the bookmark and switch the data source, it figures out which column is which (the DBs have the same schemas), but the hierarchical relationship is lost, which means I still have to manually create the hierarchies in the new version if it's going to be useful to use as a source of reports.
I realize I'm being slightly oddball in the "swapping backend DB server" use case, but I think anybody who has a data warehouse with conformed dimensions is going to want to create hierarchies on the dimensions which get re-used across fact tables, as in my original version of this question. It's great how Tableau makes it easy to work with a possibly unstructured data source; it would be nice if it were similarly rewarding to people who put time+effort into the data warehouse design beforehand!