Gravity theories with the equivalence principle but different from GR Einstein's general relativity assumes the equivalence of acceleration and gravitation. Is there a general class of gravity theories that have this property but disagree with general relativity? Will such theories automatically satisfy any of the tests of general relativity such as the precession of mercury or the bending of light?
 A: Dear Carl, my overall moral answer is the opposite one to the first two answers, so let me write a separate answer. My overall message is that with the right minimal extra assumptions, a correct theory respecting the equivalence principle has to agree with GR at cosmological distances.
There are various modifications or competitors of GR, see e.g. this list:

http://en.wikipedia.org/wiki/Classical_theories_of_gravitation#Articles_on_specific_classical_field_theories_of_gravitation

First of all, most of the theories tend to be given by an action. This is needed to preserve the Noether conservation laws for symmetries, and so on. No interesting (viable enough and new enough) non-action theory is known, as far as I know.
The equivalence principle requires that local physics must know how it's mapped to the flat Minkowski space - so there must exist the metric tensor at each point. However, there may also exist other fields or degrees of freedom.
One class you find in the list above is Brans-Dicke theory. It's an example of a broader class of theories that contain new scalar or tensor fields, besides the metric tensor. Torsion is a popular, frequently discussed addition by the people working on "competition to GR".
If such fields are massless, they may modify the long-distance physics. But they effectively destroy the equivalence principle, too. If the extra fields are massless (like moduli, massless scalar fields), their values should be viewed as a part of the gravitational field but these values influence local physics. That's really forbidden.
So when understood sufficiently strictly, only the metric tensor should form the spectrum of massless (or light) fields.
Then, you may have many theories which are GR plus additional fields. The additional fields are then treated "matter fields". They can be anything. An interesting subclass are Kaluza-Klein theories. In them, many new massive fields may be understood as coming from a higher-dimensional general relativity.
With the equivalence principle fully respected, we really deal with theories defined by an action where the metric tensor is the only massive "gravitational" field. The action for the other fields - matter - is a separate question and there are of course many choices. The action for the metric tensor must be diffeomorphism invariant.
One may prove that such an action, to have the invariance, must be a function of the Riemann tensor and its covariant derivatives, with properly contracted indices. For example, terms like 
$$\nabla_\alpha R_{\beta\gamma\delta\epsilon} \nabla^\alpha R^{\beta\gamma\delta\epsilon}$$
are tolerable. However, all such terms contain either extra covariant derivatives or extra powers of the Riemann tensor, relatively to the Ricci scalar $R$. For dimensional reasons, all such extra terms have to be multiplied by a positive power of a distance $L$ and it seems likely that all such distances $L$ that could occur are microscopic or ultramicroscopic. So all such terms become negligible at cosmological (or astrophysical) distances. In this long-distance limit, the gravitational theory has to be given by the Einstein-Hilbert action e.g. Einstein's equations and the predictions for Mercury and bending of light are inevitable.
It wasn't an accident that Einstein ended up with the right theory. He had to - and many of his "heuristic" arguments (especially "minimality") may be justified by the more valid RG-like $L$-argument above.
A: The real question here is the meaning of the Equivalence Principle: i like to think of in terms of the Fröbenius Theorem.
In this sense, you could modify the Action by powers of the scalar curvature (Ricci scalar) and get equations of motion different than those of GR — but the Equivalence Principle would still be satisfied (to some approximation). See, for example, $f(R)$ Gravity.
A: I would like to pick up one class of theories touched apon by Lubos, namely the Torsion extensions of General Relativity. The classic one being the Einstein-Cartan Theory.
This theory introduces some form of spin coupling in addition to (metric) gravity. The Torsion tensor is zero in GR and this theory maps Torsion onto a spin connection (both antisymmetric). All this will also cause the Stress-Energy Tensor to not be symmetric (unlike in GR). Note that these spin components are somewhat classical concepts.
So strictly this theory does not obey the Equivalence Principle (I think) as "spin" components could result in a different measure in a gravitational field. However the Wikipedia article states that any deviation is at $10^{-15}$ and so undetectable (at least on Earth). Some quoted arguments point out that the theory is valid around rotating Black Holes, so effects might be noticable there. Otherwise the Einstein-Cartan theory is not ruled out by observations.
Wikipedia also tells us that these ideas formed part of the basis of Loop Quantum Gravity (Ashtekar Variables); and introducing Torsion does provide some element of a mathematical completion to GR. (I believe that Einstein was not aware of the Torsion concept in 1918 and so it is zero in GR simply because he didnt know of it, rather than that it was explicitly considered and rejected. Once Einstein learned of Torsion he became excited by it and even wrote popular News articles explaining it.)
A: While it is a trivial example, any set of equations in the form of the GR equations with different constants would satisfy the equivalence principle while not being confirmed experimentally.  For example, if one took G to be 42 (in conventional units) and the speed of light to be 200 meters per hour, you would have a logically consistent set of equations that met the equivalence principle but did not reproduce GR.
A less trivial example is that the value of the cosmological constant is not constrained tightly by anything else in GR.  Other than Hubble constant scale astronomy observations, changing it really doesn't change anything else in GR.
Bekenstein's TeVeS theory is an example of a non-GR set of equations that satisfy the equivalence principle while producing effects similar to those observed an attributed to dark matter in the weak field that does reproduce the classic GR experimental tests.  (To be clear, I am not stating that TeVeS is the correct theory, particularly after the Bullet Cluster, merely that it has the mathematical character we are discussing.)  More generally, Bekenstein's effort suggests that it is possible as a general matter to craft equations that reduce to GR in most cases and meet the equivalence principle, but differ from it in a systemic way in either the strong field case, or the weak field case, or both.
Similarly, there is no obvious reason in my mind that one couldn't have a set of equivalence principle compliant, non-GR equations in which gravity was also, for example, a function of overall electro-weak charge, which is neutral in all known cases where GR is observed in astronomy.
A: Polarizable vacuum representation of general relativity by Puthoff, 1999
Abstract:
Standard pedagogy treats topics in general relativity (GR) in terms of tensor
formulations in curved space-time.  Although mathematically straightforward,
the curved space-time approach can seem abstruse to beginning students due
to the degree of mathematical sophistication required.  As a heuristic tool to
provide insight into what is meant by a curved metric, we present a
polarizable-vacuum (PV) representation of GR derived from a model by Dicke
and related to the  "THεµ"  formalism used in comparative studies of
gravitational theories
pag 11 : Comparison of the Metric Tensors in the GR and PV Approaches

Having shown by specific calculation that the
PV approach to the three classical
tests of GR reproduces the traditional
GR results

(the gravitational redshift, the bending of light and the advance of the perihelion of Mercury)
A: The principle of equivalence is built on the observation that motion of matter due to gravity is independent of any of its specific characteristics, it is universal. Contrast that with the electric force, where different particles with different charges will move along different trajectories. Gravity is different because of the equality of the gravitational and inertial masses.
Since gravity affects everything in a universal way, you can declare it to be a property of spacetime instead of being a force. In Einstein's gravity the effect of gravity is summarized in the geometry or more specifically the metric of spacetime. A priori you could imagine that gravity can be manifested in some other universal property of spacetime, but metric is the only choice I know of. 
But, once you choose the metric to encode the force on test particles due to gravity, I think the principle of equivalence doesn't give any more information. For example, it does not tell you how the metric responds to matter sources - how much spacetime curvature is created by some source of energy-momentum. There could be many answers to this question, all of which would satisfy the principle of equivalence, most of which will differ (potentially dramatically) from GR. For example, if you replace the Einstein-Hilbert action by some more general function of curvature invariants, you'd get a modification of GR which is generally covariant and satisfies the equivalence principle.
A: Since long ago there exists Relativistic Theory of Gravity (RTG) by A.A. Logunov with co-authors where gravity is a physical field in the Minkowski space-time acting on matter as an effective metric of an effective Riemann space-time. It carries energy-momentum and is different from pure geometry. In this theory there are additive conservation laws. It describes all known experimental data but it does not predict black holes (there is no singularities in it). Instead there are heavy objects of finite size. Some Logunov's papers are now available on arXiv.
A: I don't know that there's any general class of theories that compete with GR. If a competitor doesn't agree with experimental tests to date it wouldn't be viable.
The vast majority of tests of GR have been tests of the Schwarzschild metric. It's possible to tweak that metric so that it's compatible with quantum mechanics (unlike GR) yet is still confirmed by every experimental test of the Schwarzschild metric to date, including the anomalous perihelion precession of Mercury. Also the black hole information loss paradox vanishes. Tweaking the metric, as long as it's still a smooth curvature, doesn't necessarily deny compatibility with the equivalence principle.
