Even though the majority of the action took place on a planet called Third Earth, I never thought it was actually our Earth. I always thought it took place on some alien planet and the reason for having earth in the name was so kids could relate to it more. I recently saw the episode where Mumm Ra found excaliber on Third Earth which got me thinking, "Did I miss some dialog or narrative that alluded to Third Earth Being our Earth in the past or the future"?
IO9 wrote about this earlier this year, concluding that "Third Earth" is actually just our Earth, far in the future.
3) There were actually plenty of humans on the show.
While the ThunderCats and their weird animal allies and foes were obviously ThunderCats' stars, the show was not devoid of humans, who stopped by on a reasonably regular basis. There was an intergalactic big game hunter named Safari Joe (who dressed in traditional "big game hunter" garb); a variety of bounty hunters; a samurai from "the Planet of the Red Sun"; and many more. There were actually a bunch of humans already living on Third Earth as well, although 90% of them were royalty and/or wizards, like Queen Tartara and Brodo. And there was even a tribe of Amazons living on Third Earth called the Warrior Maidens (a bit on the nose, if you ask me) who became friends with the ThunderCats. Why did Third Earth have Amazons? Well, because…
4) Third Earth is actually just Earth.
Specifically, our earth is First Earth, and then it had a couple of calamities and so that the planet is now in its third… stage, I guess? But however weird Third Earth gets, what with the giant and the unicorns and the Berbils, it's still our planet in the future, hence the humans still living there, as well as the humans scattered across space. If humanity alone isn't enough to convince you, in the first episode, Mumm-Ra is shown clearly living in the great pyramid of Giza,. In another episode, the Ancient Spirits of Evil tell Mumm-Ra that Excalibur was the greatest sword ever made, so Mumm-Ra disguises himself as King Arthur and borrows it from the Lady of the Lake. So unless the Lady of the Lake has a spaceship and map of the galaxy's best lakes, it was probably the original. Also, Merlin has to stop by to save the day, and if he had to travel through time and space to get there, he didn't mention it.
Within the show itself, the next closest thing we have to direct evidence is this bit from the episode The Unholy Alliance
Mumm-Ra: As long as evil exists, Mumm-Ra lives!
Slithe: What manner of being are you?
Mumm-Ra: You need only know that I am Mumm-Ra and that I know of your mission here on Third Earth. You seek to possess the Eye of Thunder.
Slithe: You know of the Eye of Thundera? Yessss?
Mumm-Ra: I have known of its power for a thousand years! From a time when this was still First Earth!
So, there is no 100% confirmed evidence that "Third Earth" is our Earth, but there is enough evidence to make a reasonable argument.
Leonard Starr states in the below interview that Third Earth is Earth after 2 different nuclear wars.
Jules had wanted a team so I added the other Tcats, their specific weapons, Snarf for comedy relief, named them, also the bad guys, Slythe, Monkian, Jackalman, etc, called them Mutants because it was a word that resonated with kids as bad guys… like the Tcats weren't mutants, one critic later wanted to know?… Mumm-Ra the Immortal…no, the Everliving because "Immortal" had a benevolent sound to it, anyway, as the overarching evil that awaited them on Third Earth, which was what remained of Earth after two nuclear holocausts. I sent it in before the PO closed, and heard from Jules the next day, saying they liked it so much they were even going to pay me for it and could I come in the next day.
One more thing to consider: In the very first episode they clearly navigate towards Milky Way (seen on Panthro's screen), they talk about a "blue planet" (that's what we call Earth to this day) and the most obvious: They show our solar system on the screens and plan to land on the "third planet" from the sun which is actually what astro-physicists call ours: Sol 3 (from "solar" and "third from the sun").