You have to add more than one dimension, in general. Mathematicians have studied in great detail the question of how many extra dimensions you need in order to embed a curved manifold in a flat one. One key result is the Nash Embedding Theorem, which says that you can isometrically embed an $m$-dimensional Riemannian manifold in $n$-dimensional flat space, for some $n\le m(m+1)(3m+11)/2$. (Isometric embedding means embedding in a way that preserves lengths, etc., which is what seems to be relevant here.)
That's 120 dimensions for 3-dimensional space!
This theorem only applies to Riemannian manifolds, not Lorentzian ones -- that is, it applies to space, not spacetime with its annoying minus sign in the metric. If it did apply to spacetime, so that we could apply the result with $m=4$, you'd need 230 dimensions.
As far as I know, there's not a comparably clean result for Lorentzian spacetime. There are bunches of references here. But it certainly can't be any easier to embed spacetime than to embed space!
Anyway, I think this illustrates why people don't like to think of general relativity in terms of embeddings in higher-dimensional space. It's going to be way, way harder than the standard approach.
In addition, many people have a philosophical preference not to populate a theory with unobservable entities. If you don't need those extra dimensions, why postulate them?
One addendum, after thinking about Marek's comment that one might expect to get by generically with only one (or maybe a few) extra dimensions, rather than the large number in the Nash theorem. I mentioned in the comments that my intuition was different, although I wasn't sure. I just want to expand on that a bit.
You can run into trouble when using two-dimensional intuition to make guesses about higher-dimensional manifolds. In two dimensions, the Riemann curvature tensor only has one component -- that is, the curvature is described by a single number at each point. I conjecture that that's the reason why it seems intuitive that you can embed 2 dimensions in 3: you just need to "bend" the surface one way at each point to account for the curvature. (Even so, it turns out that you can't embed 2 dimensions in 3, even in relatively simple cases.) But the number of components of the curvature tensor grows rapidly with dimension. In 3 dimensions, there are 6 components, and in 4 there are 20. It's wildly implausible, to me, that you could "usually" account for all of those extra degrees of freedom with one or two extra dimensions.
(A bit of a digression, just because I think it's cool: Another example of how 2-D intuition can be a bad guide to higher dimensions. The problem of topologically classifying 2-D manifolds was solved ages ago. One might have guessed that the problem of classifying 3-D manifolds would be similar, but it turns out to be vastly harder. Last time I checked, it was thought that this problem had been solved, but there was some doubt about whether the solution was correct. And in 4-D or more, the problem is apparently known to be undecidable!)
One more point: even if it's true that you can "usually" get by with fewer dimensions, I'm not sure how relevant that is. Any manifold is a possible solution to Einstein's equation (for some stress tensor), so if you try to recast the theory in terms of extra dimensions, you'll need enough extra dimensions to account for all possibilities, not just the simple ones.