Well, copying everything definitely works and gives you access to your old pilots and stuff (both needed and spare), but if you wanna be more selective or, so to speak, "start from a clean table"...
For basic retail FreeSpace2 functionality, you only need to copy the VP's in the root freespace2 directory (although in some cases you'll need to manually create the data directory, data/players/single and /multi subfolders because Linux is sometimes a ***** with making new directories and getting their permissions correctly...). The VP's are root_fs2, smarty_fs2, sparky_fs2, sparky_hi_fs2, stu_fs2, tango1_fs2, tango2_fs2, tango3_fs2 and warble_fs2.
Obviously, any mods (like mediavp's) you want to use will need to be placed on the freespace2 directory as well.
But to answer the question, the VP's really are [AFAIK] the only files you
absolutely need for making a functional FreeSpace2 installation... that, and the executable build(s) of course.
If you have a lot of customized stuff that you want to retain, then the easiest way is of course to
$ sudo cp -R "/media/hdd/WINDOZE-PARTITION/Program Files/FreeSpace2/" "/path/to/freespace2/"Substitute the paths with whatever they are in your computer, but the basic idea is the same.
AS to where you want to put them in your filesystem, that's basically your call, basically you could run it from /the/single/most/awesomest/game/in/existence/ -folder but using that path could become a bit tedious after a while, so perhaps /games/freespace2 would do the trick. Slam the VP files, possibly data directory and whatever mods you wanna run into that directory, paste a fs2_open linux build onto the directory, fix permissions, create startup scripts if you wish, and knock yourself out (and hope everything works fine).
I'm getting segmentation errors when I try to play Beyond the Red Line on Linux, for example... and sometimes it jams the X server so I need to restart it (might be a good idea to play in separate X server), and sometimes it freezes the whole box so that no keyboard command works and I need to hardware-boot (cut the power) before it goes away...
So it's kinda exactly like in Windows, but on a free OS.

...actually, not. I don't remember when FS2_Open last actually crashed on Windows on itself...