What would be the implications for current theories if gravitational waves are not detected? Let's assume that scientists trying to detect gravitational waves get a huge raise in funding, design even better experiments, and run them for decades, but just can't find any gravitational waves whatsoever. How would theories be changed to cope with this?
We're pretty confident that gravity propagates at the speed of light. So what other explanation would there be?
 A: In many ways it is almost inconceivable that they don't exist in some form or other as it would be difficult to reconcile the absence of gravitational waves with special relativity. It's only when you have an infinite speed of propagation, such as in Newtonian gravity, that you would not expect to see gravitational waves, however infinite propagation speeds are pretty much impossible to square with special relativity (the empirical evidence for special relativity is so vast, that any rival theory would have to almost exactly re-create all the predictions of SR).
It is possible that general relativity is not the correct relativistic theory of gravity, but even if that wasn't the case you'd expect very similar predictions to GR in weak fields, including similar predictions for gravitational waves.
Unfortunately gravitational waves are very difficult to detect directly by their own nature and the lack of detection, where we might expect it, points more to our lack of understanding about the physical situations in which they are produced than our lack of understanding about the waves themselves.
A: Some would say that the existence of gravitational waves has already been proved because the period of the Hulse-Taylor pulsar decreases in time in quantitative agreement with the GR calculation.  If Advanced LIGO does not detect the expected few events over the next year, the head scratching will likely involve the ability of LIGO to detect the gravitational waves.  There are a set of coordinates (Kip Thorne’s Local Lorentz frame) in which GR predicts the arms of LIGO are strained (stretched/squeezed or parallel-pipeded) and Kip says the metric remains unchanged at diag(-1,1,1,1).  "Because the speed of light is a constant in all reference frames"  (equivalent to the metric remains diag(-1,1,1,1)), a laser wave front takes longer to traverse the stretched arm and less time to traverse the squeezed arm. Therefore, the interference of the light from along the two paths changes.  So, where is the bug in this argument, and, as you asked, how would theories change if there are no detections?
Perhaps the bug in the argument is that the speed of light is NOT constant in a gravitationally strained frame (when viewed from outside the region of strain).  We know that the speed of light is constant when viewed from frames related by a Lorentz transformation (rotations and boosts which are space-time parallel-piped strains) which leave the metric diag(-1,1,1,1) invariant, but not constant in all frames.  We know the speed of light slows down when we look into a space-time strained region (the Schwarzschild metric).  This is known as the Shapiro delay for radar pulses passing near the sun and reflected from Venus.  Gravitational waves are space-space strains for which the speed of the laser light might also be different, faster along the lengthened arm and slower along the shortened arm.  This would make a Michelson Interferometer invariant under all GL(4) transformations and not just the Lorentz subgroup.  LIGO could be viewed as an extension of the original Michelson-Morley experiment null result.  A null result from a sufficiently sensitive Advanced LIGO could indicate that gravitational waves were a usual transformation between frames that changed both the coordinates and the metric leaving $ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$ invariant, just like rotations, boosts, and Schwarzschild space-time strains do.  It is unclear to me if this would be a profound result or just a tweak in an argument applying GR.
What sort of detector could then detect gravitational waves?  It must somehow involve looking from one frame into another.  An observer outside the region of gravity wave strain does see a meter stick stretched or squeezed wrt his own.
Update (2/15/2016):  LIGO has announced the detection of a pulse of gravitational radiation.  If this result stands, then Kip Thorne was correct in only straining the coordinates or the metric (but not both) for a gravitational wave, and in the Local Lorentz frame the speed of light remains unchanged along the two arms.  A gravitational wave then does not do a usual frame transformation, while a static Schwarzschild field does seem to do one.
A: There could be some noisy background that disturbs the wavelengths we seek to see 'clouds'. So in effect the gravitational waves are produced, but our LIGO, etc telescopes can't see them because there are 'clouds' in the way. These 'clouds' would be some sort of mechanism that absorbs or randomizes the gravitational waves on their way from source to our planet. 
Its difficult to come up with a form of 'cloud' that would affect these lower frequency gravitational waves, though. http://arxiv.org/abs/1401.7251 is an example of a paper that mentions LIGO and possible problems with gravitational wave transmission (although this paper does not find problems with transmission, rather the opposite). 
