According to General Relativity, Does The Past "Exist"? I'm curious about just what is meant by time being another dimension, like the three (observable) spatial dimensions. Does this imply, according to General Relativity, that the past and the future already "exist" and that we're just moving through it, as we move through spatial dimensions? Or, is time being a dimension just a mathematical construct allowing us to calculate time dilation effects? Do we even know the answer to this question?
Intuitively, it would seem to me that the past doesn't "exist" except in our memories and that the future "hasn't happened yet" so that all that exists is the present...
Clarification
I suppose I should clarify what I mean by "exists". The three spatial dimensions definitely exist. You can travel through them and they are not just a bit of abstract mathematics.
What you can't do is travel faster than C through space. But, if you could you know when you stop, there would be something there.
Now, if you were to take the hypothetical example of time travel, there are two distinct questions here that follow this same line of reasoning. For the present, we don't have the technology to travel through time and it seems like an extremely difficult thing to do. But if you could would there be something there to travel to?
So, even if it is impossible to travel through time, our theories could still inform us whether the past and future exist in the same sense as the present does along the above lines. This was not supposed to be a time travel question... Just a question about the fundamental nature of spacetime: all "presents" existing, or only one "present", and what our current theories tell us or don't tell us about the answer to that question.
Further Clarification
So, as one of the answer's helpfully pointed out, Relativity certainly states there is no "privileged present", or a present that nature prefers and that all other points in time are measured against. I understood this before I asked the question.
But, you could still ask the question of whether in an observer's own frame of reference, does his own past and future exist?
To illustrate: Imagine instead of three spatial dimensions, we had only two. When space would stretch, it would expand or contract a grid drawn on the "paper" universe's spacetime. But, the "paper" would not move out of it's plane (NOT like a trampoline).
Time would be the third dimension, with infinitesimal space sheets stacked one after another. A stretching of time would mean pulling the sheet in or out like a trampoline.
So, in this illustrative universe, if the past and future exist, there are a nearly infinite number of sheets of present slices in array from the universe's birth till it's death. If only the present exists, there is only ever one slice and it changes over time.
Can we say which view is correct in a concrete way, not just in terms of the math? (If you haven't noticed, I'm not a math guy, but I am a picture guy)
 A: 
so that all that exists is the present...

That view is called Presentism and it is clearly in conflict with both SR and GR.
In Special Relativity, the past and future "exist" simply because of the relativity of simultaneity.
Consider two relatively moving reference frames in the standard configuration and the event where both origins are co-located.
Note that all other events that are simultaneous with that event in one frame are either in the past or the future of that event in the other frame.  
To put it another way, there is no privileged "present" (spacelike hyperslice)
A: General relativity doesn't make any assertions about existence. So the answer to your question is no. 
General relativity treats time as a dimension like space (though with a different metric sign). This just means that observers with different speeds get different 3-dimensional perspective views of the same 4 dimensions, just as 2-dimensional perspective views for the same 3-dimensional object may differ.
However, general relativity is both compatible with (i) a view that spacetime grows along a spacelike hypersurface containing all ''now''s of different observers, as with (ii) a view that space-time events are predetermined and observers just move through it. The predictions of general relativity depend on that in no way.
(i) is compatible because there is a consistent dynamics from moving between an arbitrary spacelike hypersurface to one lying in the future of it, which is fully covariant. (Ths gives Wheeler's multifingered time.)
(ii) is sometimes called the block universe and is obviously covariant.
GR assumes certain equations and derives consequences of them. The equations have meaning both for those who think (0) that nothing exists except our knowledge of things (epistemic view), for those who think (i) that 4D spacetime exists laid out once and forever and we are dragged along worldlines, and for those who think that (ii) the 4D spacetime grows as we pass along world lines, and for those who think (iii) that the past does not exist but is encoded in the present as memory. None of these is essential for GR: Mathematical consistency only requires that the dynamical equations of GR are logically meaningful. They are.This has no bearing at all on the meaning of GR for reality. 
(iii) is our everyday experience - the past recedes into memory; but this is not easy to make formally rigorous. ''memory'' is not the ''present'' in the sense of a Cauchy surface - as in the answer given here by Alfred Centauri -, but what is accessible to an observer at a given time in a given position, i.e., the fields in a neighborhood of a space-time point. This is all an observer (even a collective such as all physicists on Earth) can ever know about the past, the present, and the future.
Note that whenever we interpret general relativity we have to introduce observers, and then things become noncovariant. What an observer considers to be real is always observer dependent. (What an amoeba takes as real is very different from what a child takes as real, and what a trained physicists takes as real. There is no objective notion of experienced reality. Subjective experience must be frame-dependent, as we all experience forces and velocities that have no frame-independent meaning. Even measurable information is made frame-independent (and thus comparable with theory) only by post-processing the frame-dependent raw data. 
A: You do not really need general relativity or even special relativity to get into this problem. Consider for example a generic form for Newton's second law,
$$m\frac{d^2x}{dt^2}=F(x,\dot{x}),$$
where $x$ can be the one-dimensional position of a particle on a line or some kind of longer vector possibly involving multiple particles. Then you can see the differential equation as describing the dynamics of some point in $n$-dimensional space, or you can see it as describing the geometrical structure of a line in $(n+1)$-dimensional spacetime. General relativity is just an extrapolated version of this second view, where the geometrical objects in play are quite a bit more complicated, but the basic outlook is the same.
In a sense, whether time "exists" is not really a question that any theory can answer. What is a mathematical construct and what is actually there? (and what do you mean by "there"?) This isn't something you can look to the mathematics for help.
A: General relativity, like all physical theories, is a mathematical model of the universe. By this I mean that it allows me to take some initial conditions and calculate what will happen as I move forward (or back) in time.
Speaking as a Platonist I consider that in my mathematical model the past and future exist, and the time I experience is basically just a convenient parameterisation of a trajectory through some configuration space. Actually I'd go further than this and say the configuration space exists as well.
But I'd guess you're asking about the real world, and the answer is that I don't know because I don't know whether GR and the real world are the same thing, or whether GR is just a model. You would have to go and ask a philosopher.
A: In my view, special relativity already has this "problem". 
In newtonian theory time is absolute, so everyone share past and future with everybody. The concept of past and future is clear. They can or they cannot be predetermined, there is nothing in the theory to decide on this issue.
In special relativity, you have the relativity of simultaneity. 

In the reference frame of the boosted observer, the present is the tilted line, while the static observer has the x axis as present. Therefore the present of boosted observer contains even past and future of the static observer. It's like if the future of the static observer "is already there", in the present that the boosted observer is living.
The point is that the present of an observer is a spacelike slice. In some sense, it's questionable to say that "that present" exist, indeed without superluminal speed it's impossible for any observer to go out of the light cone an see the present. Defining existence with "possible to experience", I would conclude that no, the future it's not already existing and my past it's not surfed by future observers.
Anyway it's a really nice question to ask. 
A: General relativity uses a static spacetime as a model for our universe, and it is thus natural to think that both present and future really exist. Even though topologies which allow time travel via wormholes are possible in principle, the framework does not include any sort of meta-time, ie the geometry of spacetime as a whole - and thus the energy-momentum of all matter - stays fixed, which forbids any interesting modes of time travel.
I'd also like to address some individual points of your clarification:

The three spatial dimensions definitely exist. You can travel through them and they are not just a bit of abstract mathematics.

You do not move through space alone, but through spacetime: Each second, you travel $3\cdot10^8m$ into your future.

What you can't do is travel faster than C through space. But, if you could you know when you stop, there would be something there.

Travelling faster than $c$ also means that you'd end up in someone's past. If this is your criterion, then the past does indeed exist.

For the present, we don't have the technology to travel through time and it seems like an extremely difficult thing to do.

It's not: you're already doing it right now.
It might be helpful to think about it like this: You move through spacetime with uniform velocity $c$. When accelerating, all you do is change the direction you're moving in. Because special relativity forbids accelerating from $v<c$ to $v>c$ (where $v$ would be a relative velocity in a spatial direction), you cannot move backwards in time by 'turning around'.
However, there's particle-antiparticle annihilation, which looks like Compton scattering turned right-angle in Minkowski space, ie instead of having an electron and a positron annihilate into two photons, we can interpret it as a single electron which gets bounced backwards in time.
If you take this interpretation at face value, this would mean that when you hold your hand into an antiproton beam and an interaction happens, that antiproton is a part of you moving backwards in time.
This does not mean that a human constructed from antimatter would experience time flowing backwards, as the laws of physics are (mostly) the same for matter and antimatter and the consciousness of your antimatter-self should experience time the same way you do.
The question about the subjective arrow of time can't really be answered by relativity, but the example above (holding your hand into a proton beam) actually gives a hint: In contrast to the clean electron-positron annihilation, a messy proton-antiproton annihilation creates a whole bunch of new particles, and reversing the process would mean having to recreate a multi-particle collision.
A: General relativity by itself is neutral on this issue. The conventional metric of spacetime treatment seems to support eternalism aka block universe, but the equivalent ADM formalism seems to support presentism.
A: General relativity is best understood in terms of Einstein's field equations:
$$R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}$$
The solutions to this family of equations are metric tensor fields:
$$g_{\mu \nu}$$
This is sometimes reduced to metric tensor, or metric in discussions (and is arguably incorrect). In any case, the metric tensor field assign a metric tensor to each point in space and time (spacetime) and tells us a result when we compare to vectors in spacetime.
This metric tensor field is by itself static, so it is defined for all time.  Most research questions regarding general relativity are really about trying to find out what the actual universal metric tensor field is.  
If general relativity was the full answer, then the curvature of space is defined for all time. Since energy and curvature are related, the solutions to the Einstein field equations describe the distribution of matter for all time.  There are iterative approaches to finding solutions to the equations, and Einstein himself used such approaches in the calculations for the perihelion shift in Mercury around 1913.
Basically, time is coordinate in a vector space of four dimensions.  If general relativity is correct, there is some solution to the Einstein field equations that tells us how to compare to vectors.  Those results change with the curvature of the manifold, which also inform us about the distribution of energy throughout the universe.  So it isn't so much of the past, present and future "existing" but rather these things are defined and have clear meaning in general relativity.
A: I think it is MUCH more difficult to know the future than it is to know the past. The past has already occurred, so the light emitted from any event in the past is out in space right now. For example, if you could travel faster than light, you could catch up to the light emitted from the assassination of Kennedy and sort of experience it all over again.
You could conceivably know the future if and only if you precisely and accurately know the current state of ALL things. For example, weather prediction is limited by our lack of knowledge about every single particle and condition in the atmosphere. Assuming we knew ALL things about ALL that affects weather AND we knew ALL about the current state of everything involved in the weather, we could know the weather precisely and accurately 500 years into the future.
I think it's theoretically possible but impossible to achieve. The energy requirements of gathering all the information required would be astronomical...even to predict the weather 100 days in advance. You would have to know the position and momentum of every single particle in the atmosphere. I believe it was Heisenberg who suggested that's impossible.
A: Hermann Weyl wrote:

The world simply is, it does not happen. Only to the gaze of my consciousness, crawling upward along the lifeline of my body, does a section of this world come to life, as a fleeting image in space which continuously changes in time.

When he writes "is" he doesn't mean that the world exists in time (it's always tricky to define time without actually referring to it), but that it is (non-temporal) simply there. To Weyl (and likewise to Einstein and many others), time only exists in the gaze of his (our) consciousness.
A nice analogy is that of the hurdy-gurdy. The in advance written music roll can be compared to the world, we can be compared to the hurdy-gurdy that moves relative to the music roll and the music can be compared to time that comes alive when we hear the music (of course is the playing of a hurdy-gurdy also part of the world that, accordingly to Weyl, simply is, but nonetheless it can serve as an analogy).
On the other hand, one can just as well say that it is the music roll (which in this case writes itself, or better said, is written on the spot by music roll writers during the rolling) that moves relative to us (the hurdy-gurdy) and the music exists regardless if we listen or not. That is to say, the objective music as air vibrations. The perceived music off course doesn't exist if we don't listen.
I think one can't prove which of the two equivalent interpretations is true (see the more mathematical answer by Emilio Pisanty). It's a matter of taste and I think that people who think that everything is written in advance (I certainly not belong to them) prefer the first interpretation, while people who don't think like that (I certainly belong to them but not on the grounds of the academic "free will") prefer the second interpretation. 
