Are you launching DF by double-clicking on the dwarfort.exe file from a Nautilus window? If so, I think I know what's going on.
Apparently Dwarf Fortress just assumes that the working (current) directory is set to the DF directory. This is a safe enough assumption in Windows, because if the working directory hasn't been explicitly set by a shortcut or by launching from the CLI, then the Windows binary loader automatically sets it to the image path. Wine? Not so much. It probably just defaults to your home directory.
Long story short, DF can't find the resources it needs (most notably the font files for displaying text) because it's looking in the wrong place. Launch a shell and
cd into the DF folder (probably
~/.wine/drive_c/whatever/) and then run
wine dwarfort.exe from there.
If that works, you could create a shell script to do this for you in the future. Make a file called "dwarfort.sh" (or whatever you'd like) and edit it to:
#!/bin/sh
cd ~/.wine/drive_c/Program\ Files/Dwarf\ Fortress/
wine dwarfort.exe
You'll need to edit the path according to your own system, of course. Also remember to set the shell script to be executable. You can then launch DF by double-clicking the script instead of the dwarfort executable. You can either save the script in your desktop folder, or make a symlink to it from your desktop. If you want to get fancy, you can even make a Nautilus desktop launcher that runs the shell script instead. Useful if you'd like to set an icon for it -- I borrowed the one from the Mike's Mayday distribution.
There may be a more elegant way to do this with a command-line option for Wine, but a quick scan of the man entry wasn't terribly revealing. I only recently switched to Ubuntu myself, so it's entirely possible that my solution is a completely retarded way to handle this