I am on Emacs 24.3.1. I installed the MELPA version of plantuml-mode via
M-x package-install RET plantuml-mode RET<
After a bit more digging I found the solution to this problem: plantuml-mode won't run unless the variable plantuml-jar-path is set to a non-nil value. Fairly obvious in hindsight, but still easy to get bit by since it isn't mentioned explicitly (at least not anywhere that I could find). For best results, the variable should point to the location of the plantuml.jar file in your file system. In my case:
(setq plantuml-jar-path "/opt/plantuml/plantuml.jar")
Hope this helps someone else at some point.