The simple Newton-like explanation of dipole gravitational radiation unexistence is following.
The gravitational analog of electric dipole moment is
$$
\mathbf d = \sum_{\text{particles}}m_{p}\mathbf r_{p}
$$
The first time derivative
$$
\dot{\mathbf d} =\sum_{\text{particles}}\mathbf p_{p},
$$
while the second one is
$$
\ddot{\mathbf d} = \sum_{\text{particles}}\dot{\mathbf p}_{p} = 0,
$$
indeed due to momentum conservation.
"Magnetic" dipole gravitational radiation is analogically impossible due to conservation law of angular momentum. Indeed, since by the definition it is the sum of cross products of position of point on corresponding current:
$$
\mathbf{M} = \sum_{\text{particles}}\mathbf r_{p}\times m_{p}\mathbf{p}_{p} = \sum_{\text{particles}}\mathbf{J}_{p} \Rightarrow \dot{\mathbf M} = \ddot{\mathbf M} = 0
$$
What's about general relativity? As you know, the propagation of gravitational waves is described by linearized Einstein equations for perturbed metric $h_{\mu \nu}$, and in this limit they coincide with EOM for helicity 2 massless particles in the presence of stress-energy pseudotensor $\tau_{\mu \nu}$:
$$
\square h_{\mu \nu} = -16 \pi \tau_{\mu \nu}, \quad \partial_{\mu}h^{\mu \nu} = 0, \quad \partial_{\mu}\tau^{\mu \nu} = 0
$$
Since $\tau^{\mu \nu}$ is conserved, this protects $h_{\mu \nu}$ from the contributions from monopole or dipole moments of sources as well as from additional helicities.
Formally the deep difference between gravitational and EM radiations is that we associate General relativity symmetry $g_{\mu \nu} \to g_{\mu \nu} + D_{(\mu}\epsilon_{\nu )}$ (it is infinitesimal version of $g_{\mu \nu}(x)$ transformation under $x \to x + \epsilon$ transformation) with covariant stress-energy tensor conservation (indeed, tensor current conservation, from which we can extract conservation of 4-momentum vector current), while EM gauge symmetry is associated with vector current conservation (from which we can extract the conservation of electrical charge scalar quantity). So that corresponding conservation laws affect on different quantities; the nature of radiation in EM and GR cases are different, and the first one rules primarily by Maxwell equations (and hence conservation of charge plays the huge role), while the second one rules by linearized Einstein equations (and hence the momentum conservation is genuine).
For example, heuristically speaking, due to conservation of EM charge EM monopole radiation is impossible (it is expressed therough the time derivative of charge), but nothing restricts dipole moment radiation. In GR due to conservation of momentum vector, which is related to metric (an so to gravitational waves, in the sense I've shown above), dipole moment radiation is impossible.
This, indeed as anna v said in comments, is connected with the fact that EM field represents helicity-1 particles, while linearized gravitational field coincides with the field which represents helicity-2 particles. As you see, such thinking doesn't require presence of plus minus masses.
