grep -L -Z -r 'Subject: \[SPAM\]' . | xargs -0 -I{} mv {} DIR
The -Z means output with zeros (\0) after the filenames (so spaces are not used as delimeters).
xargs -0
means interpret \0 to be delimiters.
Then
-I{} mv {} DIR
means replace {} with the filenames, so you get mv filenames DIR.