If it is displaying all your items that you saw on the old computer then the playlists are there too. Your list of media and your playlists are all in the same file. If you have one you have the other.
Make sure you really are copying everything. Compare with a full library structure.
What are the iTunes library files? - http://support.apple.com/kb/HT1660
More on iTunes library files and what they do - http://en.wikipedia.org/wiki/ITunes#Media_management
What are all those iTunes files? - http://www.macworld.com/article/139974/2009/04/itunes_files.html
iTunes 9 [and later]: Understanding iTunes Media Organization - http://support.apple.com/kb/ht3847