My understanding is that gravitomagnetism is essentially the same relativistic effect as magnetism. If so, why is it that I've heard so much about magnetic monopoles, but never gravitomagnetic monopoles? Aren't each just as theoretically well-motivated as the other? In a similar vein, one always hears that "there is no independent existence of an electric or magnetic field, only the electromagnetic field" because E and B fields transform into each other under Lorentz boosts. Shouldn't the same statement apply to gravity and gravitomagnetism?
Gravitational monopoles are forbidden by the positive mass theorem--- any configuration of GR has positive mass, and therefore is an ordinary "pole" in the analogy with electromagnetism. The analogy is not very good, because the energy is always positive, unlike charge.
The reason magnetic monopoles make sense in EM is because of electric-magnetic duality, a parity violating symmetry between the electric and magnetic charge in the free Maxwell equations that is obviously broken by the fact that sources don't have magnetic charge. The gravitational field has many magnetic field analogs, the field is the Levi-Civita connection, which decomposes into lots of different nonrelativistic things in different components, but none of these can appear without an ordinary gravitational field at long distances.
But there are ways of embedding electromagnetism into GR, as Kaluza Klein showed, and then there are special solutions of GR that can be interpreted as magnetic monopoles of the electromagnetic reduction. There is also a whole industry of finding self-dual solutions to GR, where the duality property can be thought of as analogous to electric magnetic duality, because it involves the epsilon tensor. Self dual gravitational fields, and the decomposition of the SO(3,1) holonomy in GR into two SU(2)'s are both extremely important and fascinating topics which contain many exact solutions, and the inspiration for loop quantum gravity.
If you would like some insight into self-dual solutions, this is a better question than asking for gravitational analogs to magnetic monopoles, because these analogs are not there.
You don't hear about gravitomagnetic monopoles because they, unlike magnetic monopoles, cannot be well defined within the context of GR. Gravitomagnetism is only a weak field approximation of GR which isn't Lorentz-invariant, so it is not to be expected that everything from electromagnetism has an analogy in GR.
In electromagnetism, there are two ways one can have magnetic monopoles:
- Dirac strings. They don't change the theory and are unobservable in the classical electromagnetism, but can be observed quantum-mechanically by means of Aharonov–Bohm effect.
- Changing the theory to make it symmetric under dual transformation. This requires introduction of an additional potential.
Gravitational analogue of Dirac string can be shown to be observable in classical GR.
Second approach would require introduction of an equivalent to an additional potential in GR. Equivalent of the potential in GR is the metric, and equivalent of electromagnetic field are the Christoffel symbols. So, in order to introduce the gravitational analogue of magnetic monopoles in GR, something besides the metric should be included in modified Christoffel symbols, but that wouldn't make much sense, because both metric and Christoffel symbols are properties of the spacetime geometry itself.
In addition to the other responses it's also important to point out that the Maxwell-esque gravitomagnetic equations are not Lorentz invariant, unlike Maxwell's equations. This is because energy-density and the three-vector momentum density do not form a four-vector. Rather, they are some of the components of the stress-energy tensor. This is different from EM because charge-density and current-density do form a four-vector.
Birkhoff's theorem in general relativity states that in the case of spherical symmetry, the vacuum field equations have a solution, the Schwarzschild spacetime, which is unique up to its mass $m$. (The maximally extended version of the Schwarzschild spacetime does include a white hole as well as a black hole, but the white hole is also an object of mass $m$.) This tells us that general relativity doesn't have point-magnetic-masses in addition to point-masses.
As other answers have pointed out, the analogy with E&M doesn't quite work here, because in E&M, Lorentz invariance says that if you only knew about electricity, you'd be required to invent magnetism. GR, on the other hand, already incorporates Lorentz invariance, and the picture of gravity-plus-gravitomagnetism is actually an approximation that violates Lorentz invariance.