Did people ever thought of accelerating neutrons? It is shamefully charge-free. But presumably, if we can tune its velocity, a lot of interesting things can be done, right?
So, was there any proposal for obtaining high velocity neutrons? 
 A: Fast neutrons are produced as a product of most nuclear reactions. 


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*In fission reactions, the mean energy of the released neutrons is approximately 2 MeV. Since the mass of the neutron is 939 Mev/c2, their velocity is 0.065 c.

*In fusion reactions, like the deuterium-tritium reaction, the neutron gets 14 MeV, and thus its velocity is 0.17 c.


Larger energies are obtained using a technique called neutron spallation. The idea is to accelerate a light ion, like a proton or a H-, using the conventional methods (linac, synchrotron). The accelerated beam is directed to a target composed of heavy atoms (tungsten, tantalum, uranium...). As a result of the collision, high energy neutrons are created. There are several installations of this type around the world. For example, the Spallation National Source at Oak Ridge creates neutrons with velocities greater than 0.9 c.
Once you have high energy neutrons, is fairly simple to slow them down. Just make them pass through a medium like water or graphite, called moderator. Neutrons are scattered by the nuclei in the moderator and lose most of their energy. 
Some interesting Wikipedia links: Neutron temperature, Neutron moderator, Spallation
A: In fact it turns out to be very useful to generate cold, or slow, neutrons.  A neutron which has gotten close to thermal equilibrium with some material at room temperature (kinetic energy $E=kT=p^2/2m\approx\frac1{40}\rm\,eV$) has a de Broglie wavelength $\lambda = h/p \approx 2\,\text{Å}$ which is comparable to the interatomic spacing in most materials.  This means that neutrons will undergo diffraction with ordinary materials and are very useful for exploring material structure.
In fact this is about 95% of the work that goes on at a neutron source like the SNS at Oak Ridge.
Bosoneando is absolutely correct that the prompt spallation neutrons are relativistic, but no instruments at the SNS have a direct view of the spallation target; everybody gets their neutrons from various room-temperature or colder moderators.
Other neutron sources (such as the ones at HFIR and NIST) can deliver both fast (few MeV) and slow neutrons.
Decelerating neutrons is easy enough: you just put cold stuff in their way.
