Compress the folder %APPDATA%\RenPy
. It's faster and safer, this way you are not going to miss multi-persistent files and stuff.
Then log in to Linux, move the archive into your home, unzip it and rename the directory as .renpy
(yes, all lowercase and with an initial dot). If it disappears, don't panic: it's because your file manager hides files and directories starting with a dot. Just change its settings.
A similar trick should work for Unity games too, except that you are going to play them via some Wine-based software. Find the Windows folder(s) with your saves, compress it, move the archive into Linux and extract its content inside the corresponding Wine directory. On Linux, most of my saves are stored in:
$HOME/.wine/drive_c/users/[MY_USERNAME]/AppData/LocalLow
but if you use the Steam client, the path is different.
As for RPG Maker games, their save folders are usually inside the game itself. Those must be moved separately.