So I'm stubborn and lack patience... I finally managed to solve the problem. Unfortunately, I was trying out different things and didn't keep track of what I was doing exactly, but I hope the following will be useful anyway. If you still can't get FreeCiv to work, I'll do some digging in the package manager and write down what it was I installed exactly.
I installed the Livna repositories, as explained
here. I also updated the Fedora repository using a tip on macles* site (linked from a page on this very forum), but I'll be damned what the URL was. I don't think that was a vital step, though, as I think the GGZ packages are offered through Livna, rather than Fedora.
Anyway, I loaded up the package manager, searched for "freeciv", then selected the server (won't work without it), SDL-client, common files, and the regular client. Once all that was installed, I did a search for ggz. Having added the new Livna repository, a number of options appeared. After some trial and error, I selected two packages that installed missing library files (one for Gtk+, the other a general ggz file--do NOT use the development packages, etc., just the finished files). These installed rather swiftly.
Following the installation of FreeCiv, two shortcuts (icons) should have been placed into your "Games" folder, one for the FreeCiv client, the other for the server. If you click the FreeCiv client, it will automatically load the SDL-version, which is much nicer than the regular version I was used to!
If the game doesn't load, open up the terminal and type "civclient-sdl" (without the quotes). An error message should appear, telling you what the problem is. Chances are that one of those pesky ggz-libraries are still missing. If so, relaunch the package manager, do another search for "ggz" and select another package to install.
EDIT: corrected a mistake.