I just installed ArchLinux, KDE plasma-meta package, and kde-applications group which includes this JuK player.
I put some mp3 files to the program and double clicked the song name but nothing happens. Clicking play button skips to the next song in the list.

Seems like ogg files are playing well.
`mpg123` on command line plays mp3 files well.
VLC player plays them well.

I don't have Juk installed so I may not be quite correct

Juk may only use the gStreamer phonon backend and either you haven't install gStreamer or you're missing the mp3 module something like gstreamer-0_10-plugins-fluendo-mp3

VLC uses its own codecs

Though you didn't use JuK your advice solved my problem, quite correctly!

I searched the repository for 'gstreamer' and I found 'gstreamer-0.10-ffmpeg'. Installed this and opened JuK, and now I am listening to my mp3 files with this program!

Okay I once accidentally nuked my whole system so I had to reinstall the entire system.
I got to this point of JuK not playing mp3 files. This time other apps also don't play mp3.

Tested with Amarok and DeadBeeF. All failed. `amarok --debug` seems not giving any message about codecs.
I have no problem hearing sounds from websites.

I tried installing these and didn't make it work: gstreamer-ugly, gstreamer-ugly-plugins, xine-lib.

gstreamer-0.10-ffmpeg ??

https://wiki.archlinux.org/index.php/KD ... on_backend

MP3 files cannot be played when using the GStreamer Phonon backend
This can be solved by installing the GStreamer libav plugin (package gst-libav). If you still encounter problems, you can try changing the Phonon backend used by installing another such as phonon-qt4-vlc or phonon-qt5-vlc. Then, make sure the backend is preferred via System Settings > Multimedia > Phonon > Backend (tab).

Solving the problem was as simple and strange as encountering the problem, `sudo pacman -S gst-libav` just did everything. It's so strange...
Thanks for helping me, twice. (_ _)