GR doesn't really postulate anything as far as whether or not you can travel along the time axis at will like you can along the spatial axes. It models spacetime as a 4-dimensional thing that just "is", not as something "moving" through the time axis. Interpreting the time dimension as something that objects "move" through is a way of linking up the spacetime model with our everyday experience, but you can do all the math and make all the predictions in GR without ever using it.
Mathematically, the reason the time axis is different is that it has an opposite sign in the metric. This is true in SR as well; take a look at the Minkowski line element:
[tex]d\tau^2 = dt^2 - dx^2 - dy^2 - dz^2[/tex]
The [itex]dt^2[/itex] term has opposite sign from the other terms; that means the interval [itex]d\tau^2[/itex] is not positive definite. That is, the interval between two distinct points can be positive, zero, or negative. That's not possible in ordinary Euclidean geometry: there, the distance between two points can only be zero if the points are identical, and it can never be negative.
What all this means is that, in spacetime, there is something fundamentally different about a timelike interval with [itex]d\tau^2 > 0[/itex], vs. a spacelike interval with [itex]d\tau^2 < 0[/itex] or a null interval with [itex]d\tau^2 = 0[/itex]. They are three physically distinct kinds of intervals. The same is true in GR; the only difference there is that the line element can look different than the formula above, due to spacetime curvature.
But you'll notice that nowhere in any of this did I talk about anything "moving" along a curve, or through an interval. If two points are separated by a timelike interval, that means some timelike curve connects them, so some object's worldline can pass through both points. But that's just a fact about the geometry of spacetime, in the same way that the statement "the Earth's equator passes through Quito, Ecuador and Nairobi, Kenya" is a fact about the geometry of the Earth's surface. (I don't know that that's exactly a fact, btw; those cities are close to the equator but probably not exactly on it. But it illustrates what I'm getting at.) We can describe the geometry of spacetime without talking about anything "moving" in it, just as we can describe the geometry of the Earth without talking about any objects moving on it.