So we have z = f(x,y) for some function f.
Earlier, you considered cases where either x or y were some constant (let's call it c), and you were looking at the resulting equations like z = f(x,c). Here we can clearly say that z is related to x by some function. Similarly when we considered z = f(c,y) we can clearly say that z and y are related by a function.
You are now asking about the case where f(x,y)=0.
This is an implicit relationship. You aren't guranteed that there is a function directly relating x and y. And if it is exists, it's not obviously related to f like the previous cases. Is your function f a special case, like a linear function?
If it is an easy case, you can use Solve as follows:
Solve[f[x,y]==0,y]
Since you're working with a dataset rather than a function, you can just search for the points (x,y,z) where the z component is sufficiently close to 0.
Here's an example:
exData = Flatten[#, 1] &@
Table[{x, y, Sin[x + y]}, {x, 0, 6, 0.1}, {y, 0, 6, 0.1}];
nearZero = Select[exData, -0.1 < Last[#] < 0.1 &];
Of course, now all you have is a list of points. I'm not sure if that is useful for your purpose or not. It'd be hard to infer any curves from it.
You could take advantage of the implicit function theorem and try to define a curve numerically (probably a spline).
If you are only interested in getting a plot of f(x,y)==0, then you can use
ContourPlot.