Trim first 6 characters from Folder Names [AppleScript]

This code will do it for you:

set whichFile to choose file with multiple selections allowed
repeat with aFile in whichFile
    tell application "Finder"
        set filename to name of aFile
        set name of aFile to ((characters 4 thru -1 of filename) as string) --trim first 3
        --set name of whichFile to ((characters 1 thru -4 of filename) as string) --trim last 3
    end tell
end repeat

Note that stripping the last three will get rid of the extension. If that isn't what you want to happen, let me know in a comment.

For example save this file as filenames.txt:

apple
banana
orange

Then run:

a=($(<filenames.txt));for d in *;do i=0;for f in $d/*;do echo mv $f $d/${a[i++]}.jpg;done;done

Remove echo if the output looks right.

If the file names contain spaces, add IFS=$'\n'; to the start and add double quotes around the variables.