Music player on Linux without playlists, just a directory navigation and file playing

I use moc, it does have a playlist window, but you can hide it and just play from the directory listing. It's what I turned to after giving up on new Amarok.

I use quodlibet. A nice little player with comprehensive file-type support and fabulous metadata editing abilities. It has a "File System" view, which meets your criterion.

mpg123 and mpg321 are commandline MP3 players. You could associate them with MP3 files in Thunar/Nautilus/file-manager-of-choice. They take a list of files/URLs to play as arguments:

$ mpg123 /path/to/mp3s/*.mp3
# plays all mp3 files in directory /pat/to/mp3s (generally in order of filename)

$ find /path/to/mp3s -name '*.mp3' -print0 | xargs -0 mpg123
# plays all mp3 files in /path/to/mp3s and subdirectories

See man mpg123 or man mpg321 for details.

I haven't seen 1by1, but personally I use cplay when I want to both navigate and listen using the player.

For odd tracks, I have vlc associated with sound files, so I can navigate in Thunar (or whatever) and just double click on the file.