Why not try compiling from source instead of using the latest .deb file from the main site? Those builds are built for Ubuntu (13.04, 64-bit); now I know Linux Mint is supposed to very similar to Ubuntu already, but they're still different distros that might be doing slightly different things.
Follow the instructions on the Google Code wiki entry: https://code.google.com/p/dolphin-emu/wiki/Linux_Build Tell us if you come across any issues.
Follow the instructions on the Google Code wiki entry: https://code.google.com/p/dolphin-emu/wiki/Linux_Build Tell us if you come across any issues.