Why is there no gravitational magnetic field? (Or, is there?)

There is a sort of analog called gravitomagnetism (or gravitoelectromagnetism), but it is not discussed that often because it applies only in a special case. It is an approximation of general relativity (i.e. the Einstein Field Equations) in the case where:

• The weak field limit applies.
• The correct reference frame is chosen (it's not entirely clear to me exactly what conditions the reference frame must fulfill).

In this special case, the equations of GR reduce to:

$$ \begin{align} \nabla\cdot \vec{E}_g &~=~ -4\pi G \rho_g \\[5px] \nabla\cdot \vec{B}_g &~=~ 0 \\[5px] \nabla\times \vec{E}_g &~=~ -\frac{\partial \vec{B}_g}{\partial t} \\[5px] \nabla\times \vec{B}_g &~=~ 4\left(-\frac{4\pi G}{c^2}\vec{J}_g+\frac{1}{c^2}\frac{\partial \vec{E}_g}{\partial t}\right) \end{align} $$

These are of course a close analogy to Maxwell's equations of electromagnetism.

There is a gravitational analogue of the magnetic field. See gravitoelectromagnetism and frame dragging on Wikipedia.

The reason that magnetogravitational fields don't appear in purely Newtonian gravitation is that magnetism is actually a relativistic effect. If you use the CGS system of units, you'll see that only the quantity $B/c$ appears in the Lorentz force law. The nonrelativstic (Newtonian) limit is equivalent to the limit $c \to \infty$, so in this limit the magnetic fields drop out entirely.