It turns out that phpBB has some features in v3 that aren't enabled by default, but which might help cut down the junk sign-ups. Basically, I've now enabled the "check IP against blacklist" option, so all registrations will be checked against a list of known spammers. I've also required numbers in passwords, which should hopefully cut out a few more spam bots as well as making people's passwords stronger.
If anyone notices any slow-down in posting after these changes then please give me a shout and I'll investigate.