I'm looking for a basic Linux-to-Linux video chat solution. No file transfer, desktop sharing, etc. needed.

Google+ Hangouts sounds nice, it gets nice reviews. But I'm a little wary of having to set up Google+ accounts, login to Google, allow them to set cookies, use javascript, etc. Normally I block all that stuff for Google. However, Hangouts would give cross-platform support even though I don't require that now - but who knows who (and what OS) I may want to chat with later?

The systems I am immediately concerned about video chatting between are both running LinuxMint13 with the Xfce desktop. Chances are 95% that this is all I will ever require. Webcam on system #1 is a Logitech C270 and I'm planning on buying that same webcam for system #2. This webcam has a built-in mic and is 720p. Are there any good echo-cancellation video chat clients I could use with the webcam mic and normal computer speakers, or should I just go with headsets?

I looked into this several months ago (asking here on LQ.org). At that time I was also restricted with low powered computers and needed low-resource clients. I never got around to finalizing anything back then, although I played around with Mumble/Murmur. Now, low-resource clients are no longer required (I hope!)

Thanks in advance for tips, pointers, and recommendations.

I never used one of them, but Ekiga or Jitsi may be what you are looking for.
Both are cross platform, both have Ubuntu packages available (Ekiga is in Ubuntu's repository, Jitsi offers their own repository):
http://www.ekiga.org/
https://jitsi.org/

I use Google+ for video chats all the time. It's far better than Skype or anything else I've tried. You don't have to give Google a real name, address, or anything else. You can get a free account using entirely fictional information, and I have several of those, for various purposes. You can use Ghostery in Firefox or other browsers to block trackers, and Hangouts still work fine.

The quality, both video and audio, is excellent. I've used it both with headsets, and using the speakers, both on laptops using the built-in speaker/mike/camera, and on desktops, using external camera/speaker/mike, as well as smartphone and tablets, and all are pretty much comparable. I've never noticed a problem with echo. A headset does provide better noise cancellation, if background noise might be a problem, but otherwise I haven't seen the need other than for privacy concerns, where others might hear the speakers. Barring that, just plug in whatever hardware you have and go at it.