No. Gravity in General Relativity is not just about the mass. Einstein's equation sets the curvature of spacetime equal to the Energy-Momentum-Stress tensor, which contains one term for each pair of coordinates from x, y, z, t.
The tt term is the energy, and because of the rest mass energy usually pointing in this direction when things are going slowly, this is usually by far the biggest term. That's why it is usually valid to ignore all the other terms and treat gravity as if it was purely sourced by accumulations of mass.
The xt, yt, and zt terms are the linear momentum. When things are moving very fast, these terms get big. However, because of the peculiar geometry of spacetime, it tends to cancel out the effect of the mass. More on that in a moment.
The xx, yy, and zz terms are the pressure. This is one of the main reasons why stars collapse into black holes - as the star gets bigger, the pressure in the centre grows, which makes the gravity even stronger, raising the pressure even further. There's a positive feedback loop that means beyond a certain point infinite pressure is needed to balance the forces. Since nothing can supply infinite pressure, nothing can stop the collapse.
The xy, xz, yz terms are the shear stresses. Those are usually small, because no matter is strong enough to resist such forces. With compression, there's nowhere for the matter to go, but with shear the matter flows like taffy.
If you rotate your coordinate system, the numbers change. But it's still the same underlying geometry - just described differently. Rotating about planes involving only x, y, z axes behaves like you would expect, but rotating in any plane involving time works a bit differently. In Euclidean geometry Pythagoras says the squared length of a vector is $x^2+y^2+z^2$, which rotations don't change. In Minkowski geometry, the t coordinate has the opposite sign. So we can define the squared length of a vector using a modified Pythagoras theorem as $t^2-x^2-y^2-z^2$, which doesn't change under 4D rotations. This leads to a relationship between energy and momentum: $E_0=E^2-p_x^2-p_y^2-p_z^2$, where $E_0$ is the rest energy (or mass), the length of the energy-momentum vector and a fixed constant in any reference frame, $E$ is the relativistic energy (or mass, the thing that increases with speed), and $p_x$ etc. are the components of the momentum.
When you shift to a reference frame where the object is moving, the momentum terms obviously get bigger, and so the energy term gets bigger too. This is the kinetic energy. The increase in size of one is counterbalanced by the other, to yield exactly the same rest energy. The energy-momentum vector is not any longer - it's just viewed from a coordinate system where the components in each direction are bigger numbers. And so the 'size' of the whole energy-momentum-stress tensor, and hence the gravitational space-time curvature, is no different.
This does not mean that kinetic energy is irrelevant for gravitational purposes. If you have a fast moving particle confined in a box, then it bounces backwards and forwards, resulting in momentum currents in opposite directions that cancel out. But the kinetic energy term doesn't cancel. If particles are bound together, then the kinetic energy of their confined motion contributes to their mass, because the momentum terms cancel out. It turns out that much of the mass of ordinary matter is not the bare mass of the constituent particles themselves, but comes from the binding energy that is the result of their confinement, being stuck to one another.
