Are there alternatives to General Relativity with different outcomes?

Accepted is not really the right word.

Many different tests that we are capable of performing IMPLY VERY STRONGLY that General Relativity in its most simple form, is the way that nature seems to work. This means that the underlying principles are also sound. Specific ones like the bending of light by gravitational fields and the equality between gravitational and inertial mass can be individually tested and have been found to be correct to within experimental error....one part in 10 trillion for the later test in fact. Let's have a look at how the tests and predictions are stacking up so far:

 Test Prediction Measurement Accuracy Newton Einstein Ratio of grav to inertial mass 1.00000000 1.00000000000005 5:100 trillion No Yes Total Precession of Mercury Perihelion 574.64±0.69 arcsec/century 574.10±0.65 asec/century 1:1000 No Yes Light Deflection 1.75 arcseconds 1.76+/-0.016 arcseconds 1:100 No Yes Gravitational Redshift Prediction Measurement 14:100000 No Yes Gravitational time dilation Prediction Measurement 7:100000 No Yes Frame Dragging Effect -39.2 mas/yr -37.2±7.2 mas/yr 1:20 No Yes Gravitational Radiation Prediction Measurement Accuracy No Yes Binary Pulsar Timing -2.403x10-12/td> -2.409x10-12 Accuracy No Yes

Sources: Wikipedia:Tests of General Relativity; http://einstein.stanford.edu/highlights/status1.html; http://www.physics.umd.edu/courses/Phys798g/Paik07/Lecture0208.pdf; https://en.wikipedia.org/wiki/Equivalence_principle;https://arxiv.org/pdf/1508.06159.pdf

So far, there are simply no places to sneak a completely different theory that does better than General Relativity, and this is somewhat of a problem because without signs that General Relativity is in bad shape in some way, there is no maneuvering ground for introducing a 'better' theory. There would be nothing for the new theory to explain better at the current time. This also means that General Relativity as it stands is our best shot. But perhaps in the next century some new test will be discovered so that the field can progress further.

Still..between curved space, black holes and the Big Bang, what more could you possible want to 'explain' by a better theory?