If you have a second hdd connector, I would connect the alternate hdd, and run the following commands (This is assuming that the hard drive is mounted as hdb, and that partition 3 is your /home partition); also if you have an auto-mounter then it may mount the partition for you, so you don't have to run mkdir and mount (just adjust the path in cp to point to where the auto-mounter put it):
mount /dev/hdb3 /tmp/home/
cp -r /tmp/home/ <where you want the files to go>
Please note that I just came up with the above commands off the top of my head. To the best of my knowledge they are correct; but as Donald Knuth said "I have only proved this code correct, not tried it."
Lacking a free hard drive connector, then you could put it in as your primary disk, boot off a live-cd, and copy the files from your home directory to a USB key, online data storage service, burn them to a CD, or some related transfer method.