Here is another method based on `RegionPlot[]`, similar to rm's solution. There are a few wrinkles in this version, however:
1. I use `PolarPlot[]` to generate the ticks for me. (I know about the hidden functions behind the generation of the polar ticks, but I couldn't figure how to use them directly.)
2. I use the saturation and brightness arguments of `Hue[]` to generate the meshes as part of the color function. The idea was <strike>stolen</strike> adapted from the solutions of Heike and Simon in [this answer](http://mathematica.stackexchange.com/a/7359), but I did change a few things around.
<!-- There might be a better way to do this... -->
Now, on to the routine:
hueWithMesh[x_, y_, hx_: 1/10, hr_: 1/8, ht_: 1/24, r1_: 2/5, r2_: 1/2, g_: 1/5] :=
Block[{ph = Arg[x + I y]/π, s, b},
s = r1 + (1 - r1) Abs[(Mod[2 Abs[x + I y]/hr, 2, 1] - 2) (Mod[ph/ht, 2, 1] - 2)]^g;
b = r2 + (1 - r2) Abs[(Mod[2 x/hx, 2, 1] - 2) (Mod[2 y/hx, 2, 1] - 2)]^g;
Hue[ph/2 - 1/12, s, Max[1 - s^2, b]]]
Show[PolarPlot[1/Sqrt[2], {t, -π, π}, MaxRecursion -> 0, PlotPoints -> 6,
PlotRange -> 1, PlotStyle -> None, PolarAxes -> Automatic],
RegionPlot[Abs[x + I y] <= 1, {x, -1, 1}, {y, -1, 1}, BoundaryStyle -> None,
ColorFunction -> (hueWithMesh[#1, #2] &), ColorFunctionScaling -> False,
Frame -> False, PlotPoints -> 200], PlotRange -> All]
As you might notice from the implementation of `hueWithMesh[]`, the parameters `hr`, `ht`, and `hx` all control the spacing in the rectangular and polar meshes, while `r1`, `r2`, and `g` all control the saturation/brightness for the meshes. You can tweak these parameters to your taste.