Scratch

As an alternative to Kileymeister's script, you could set the center of the sprite to a point where the planet it will be orbiting will be, then simply use a [Go to sprite] block and rotate the sprite. This also uses rotation of the sprite itself, but it's simple. I'm afraid you probably won't find a method that uses neither rotation nor trigonometry.

Building on Harakou's idea; Use a "guide" sprite to move the planet.

In the guide sprite;

Go to (Sun)
Rotate (desired angle step)
Move (desired radius)
Broadcast "Update Planet" and wait

In the Planet sprite;

When I Receive (Update Planet)
Go to (guide)

You could begin with the guide sprite visible (and in front of everything else) so that students can see what is going on (you could also have the planet "wait 0.5" after moving if desired).  After it's clear how the program works, hide the guide sprite.

This could be expanded to handle more planets, after the first planet, have the guide repeat its move with a different rotation angle and radius, the broadcast "Update Planet 2" and wait.

I've used the guide sprite idea to move a "spaceship", where the direction you are pointing and the direction of travel have to be maintained separately.  These project do use Sin and Cos; Trig sure makes it easier...

The moon phases idea is a nice project.  Another idea comes to mind, especially if you intend to do a smooth orbit;  You could replace the pictures of phases with a picture of the moon and slide a "mask" sprite over it in time with the orbit motion.  The mask would just be a black sprite with the right shape, "on top of" the moon picture.