Applications of the Linearized Einstein Field Equations (EFE) Look up linearized Einstein field equations anywhere and the first thing you'll see will be a discussion of gravitational waves. Using the linearized EFE's is pretty handy when studying gravitational waves, but it doesn't seem like they are used anywhere else! Is this true? If not, what are the other applications?
 A: 
If not, what are the other applications?

Calculating the relativistic precession of Mercury, for one. This post-diction was one of the key things that helped with the rapid acceptance of general relativity.
Modeling GPS, and calculating the orbits of LAGEOS and Gravity Probe B, for another. A full-blown general relativistic formulation works quite nicely on (and is absolutely essential for) black holes and neutron stars precisely because gravity about those extremely massive objects is simple. Earth's gravity field isn't so nice and simple. It's rather lumpy compared to a neutron star. One of the more recent models of the Earth's gravity field, Earth Gravity Model 2008 (EGM2008), is a 2159x2159 spherical harmonics model. How are you going to handle that with general relativity? The answer is to linearize the field equations. 
Modeling the behavior of the solar system, for yet another. All three of the leading models of planetary ephemerides use a first order post-Newtonian approximation of gravity. (But apparently they're starting to wonder if they need to step beyond that. To second order.)
One last use: "weigh" the Earth. See my answer to the question "How is the mass of the Earth determined?" at the earth science stackexchange sister site.
A: It could be used to account the gravitoelectric effects http://en.wikipedia.org/wiki/Gravitoelectromagnetism
These equations (similar to the maxwell equations of electromagnetism) allows explain astronomical observed phenomena in a simple way, in a linear theory. For example relativistic jets are explained with these equations.
A: I will try to approach the question from a different point of view:
Linearized Einstein equations often appear in the context of the AdS/CFT correspondence, which is a realization of the holographic principle: it relates a theory containing gravity (string theory) on a certain space (Anti-de Sitter) to a quantum field theory living on its boundary. The idea is that everything on the field theory side has a counterpart on the gravity side, i.e. there exists a so-called holographic dictionary. 
Within this framework, gravitational waves correspond to certain field theory operators on the boundary. Solving the linearized field equations allows us to extract information about the them. An example where this is useful are formulations of the duality which exhibit confinement: gravitational waves correspond to specific gauge-invariant operators of a Yang-Mills theory, i.e. to glueballs. Their mass spectrum can be determined simply by solving the linearized Einstein equations. 
Even though this still involves gravitational waves on the gravity side, it is essentially more than that: you are describing strongly coupled gauge theories!
A: Recently (in 2006) Glampedakis and Babak have used the linearised Einstein equations to derive what is called the 'quasi-Kerr' metric (http://arxiv.org/abs/gr-qc/0510057).
The idea is that in general relativity the combined uniqueness theorems of Israel, Carter, Hawking and Robinson show that an uncharged black hole must have exterior field that is Kerr (no hair theorem: http://en.wikipedia.org/wiki/No-hair_theorem). The LISA detector is a new technology coming out that seeks to map out photon trajectories (in GR they travel along geodesics). If non-Kerr features are found then it would indicate that the theory of general relativity breaks down near these massive, rotating black holes. So, LISA will give us some bounds $(\pm)$ on just how close to geodesics the photons actually move in space-time. What does this tell us about the theory of gravity?
So Glampedakis and Babak have effectively answered the question of $\mathit{\text{how}}$ different the theory would be from general relativity if deviations were found (in terms of the order of the multipole contribution). They used little other than the linearised Einstein equations to build more general `solutions' than Kerr.
So the linearised Einstein equations are still very valuable tools!
