I've done this the hard way a few times now (by hand) and finally took a moment to find instructions online how to do this. I've confirmed that it works like a charm if you use PuTTY and need to install it on a new pc while saving your connections.
(As for the original source of this, I don't know -- I found the same instructions in multiple search results)
-----------------
PuTTY stores its session information in the registry...
1. On the old computer, open up a command prompt, and run:
regedit /ea new.reg HKEY_CURRENT_USER\Software\SimonTatham\PuTTY
2. Copy new.reg onto the new computer
3. On the new computer, open up a command prompt, and run:
regedit /s new.reg