Ok, first off, as far as I am aware, TeamSpeak is a voice chat application rather than a network virtualisation one. It's not something I use often, so there may be a facet that I'm unaware of, though.
As far as running servers go, I'm assuming that you have a direct internet connection (as opposed to the more common router, or other always-on based connection), in which case you've got a few options:
1. Network virtualisation. You've used it before, there's not really much difference (from a network perspective) between vanilla and modded minecraft.
2. Dynamic ip direct connection. Assuming that your ISP doesn't block any traffic, you could set up your server on your machine, and just tell your friends the IP at the beginning of each session. It's a pain, but if you have a thread/group message, it's not beyond reason.
3. Dynamic DNS. There are servers that give you a fixed domain name (for free versions, usually a sub-domain of their primary one), and install an application which continually updates that domain with your current ip. It's a little more heavy on the setup, but if you can get it running, probably the easier route in the long run.