What kind of computers are you running off of and do you only want to transfer the playlists or the actual mp3s you purchased from your old computer to your new computer.
ON A MAC:
in iTunes on your old computer, go into the playlist you speak of, highlight one of the tracks in that playlist: Control and click one on that selected MP3. In the contextual menu that pops up, select: SHOW IN FINDER
A folder will open up with that MP3 in it. That should be the MP3 you want.
Do this to each track if they are all not by the same artist and are not in the same folder as the first MP3 you've done this to.
If you already transferred all the purchased music from old computer to new computer and just want to transfer playlist; Select the playlist you want to transfer. Go to File // Library // Export Playlist
Select a format (I believe default is Unicode Text but that M3u will work as well)
Save to a destination or a flash drive and hit ok
Take the flash drive or harddrive, pop it onto your new computer, double click the exported playlist and it should automatically open itunes and list the contents of that playlist.
Hope any of this helps.
Step one is to plug in a thumb drive or external hard drive big enough to hold all the tracks of your playlist. Once it shows up on the desktop, go to your playlist, Select All, then drag then entire bunch onto the external drive. At this point copies of the tracks will be created on the external drive.
Step two, is back in your playlist. Now go to File -> Library -> Export Playlist... In the pop up window in the pull down for Save As Type: change it from TXT format to XML format. And also point to your external drive an export it there where you already put the music. This will create an XML file of your playlist. Now plug the external into the other computer with iTunes and open iTunes. Drag and drop all the songs from the external into iTunes. This adds the media files.
Finally to recreate the playlist go to File -> Library -> Import Playlist... and pick the XML file off the external drive.
<Link Edited By Host>