One simplification of what gravitational waves are and how they're created, can be understood by visualizing a massive body and its gravitational field when stationary and what happens when the body accelerates.
Imagine what happens if such a body were to start accelerate to the right for a while, and then stop. Just like the time-delay with light, it will take time for distant objects to notice the motion / change in position of a distant gravitational source. In one sense, the motion has created a ripple effect for distant observers - on the right, observers will notice that around the time they can see this body move, the gravitational field at their location has been increased in strength; in other words, a ripple in the curvature propagated to the right that resulted in slightly more curvature for the observers on the right than before. A similar effect would occur for observers on the left with a weakening strength.
These 'ripples' so far do not need a medium for the effect to be understood. The 'medium' here, would be to have an observer at every point in between. That is, if you could observe this effect from everywhere and collect the results, you would see a propagating wave in the data despite awareness of any medium.
Any phenomena that involves significant energy densities and accelerations, like stars, black holes, galaxies or even events near the epoch of the big bang etc, would be capable of producing significant gravitational waves.
I realize this doesn't directly answer whether or not gravitational waves are due to spacetime tension or similar. Strictly speaking it's not right to think of it as tension...
If we were dealing with an electric charge and its electric field... The change in electric field due to motion, would induce a changing magnetic field (while the motion isn't constant), which in turn induces a changing electric field etc - propagation in a sense. You can think of gravitational waves as having this sort of interplay, but instead of electric charge you have mass 'charge', so the interplay is between mass-energy and curvature, each being the cause of the other.
As consequence, gravitational waves do carry mass-energy and exhibit their own gravitational field, odd as that may sound.