transfering songs to new computer using external HD

So I'm using an external hard drive to transfer songs from the iTunes on my old computer to the iTunes on my new one. All of the songs that I DOWNLOADED on my old computer have transferred just fine, but I'm having trouble with all of the music from my old computer that I ripped from my CD collection. On the new computer, the files are listed in iTunes, but when I try to play them, iTunes says it can't find them. If I plug in my external hard drive, voila! The songs start playing, but the second I turn off the hard drive, they stop working again. I have tried to save them using the "import" button, and the "add files" or "add folders" button, but nothing seems to work. Any ideas? Thanks in advance!