Update:
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There is an ongoing effort to develop the new flat icon set for LibreOffice. Please see this whiteboard wiki page for more info. The work is led by Issa Alkurtass. Icons are based on GNOME Symbolic theme and will probably be called Sifr in LibreOffice.

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Building the icon set
The icons (stencils.svg) can be downloaded from the libodesign/flat-icons repository on GitHub.

In the SVG, each icon is the group or a clone with the ID property set to the icon name and the label is the same or set to just "icon". I used a Python script and Inkscape to export individual PNG files. To build the expected directory tree, the Python script will look for the same icon name in the Galaxy icon set and use the matching path when building the new set.

At first, we will need the Galaxy icon set. From the LibreOffice installation directory extract "share/config/images.zip" to the new "galaxy" directory. Next, download the most recent "stencils.svg" from the libodesign/flat-icons repository.

Now download cnvsvg.py script and run it as: python cnvsvg.py stencils.svg flat galaxy. As said, this will use Inkscape to extract individual icons to the "flat" directory, taking "galaxy" as the reference directory tree.

The flat icon set is not complete so to fill the missing icons we will use the Galaxy set again, but will also convert icons to grayscale for a better fit using ImageMagick: 

for i in `find galaxy -name "*.png"`; do
    convert $i -channel RGBA -matte -colorspace gray $i.png;
    mv $i.png $i;
done

Copy the flat icons over modified galaxy set (cp -r flat/* galaxy/) and create the new icon set archive (pushd galaxy; zip -r ../images_flat.zip *; popd).

Finally we will add this icon set to the installed LibreOffice. As we will not recompile the LibreOffice from the source code (see here for a required patch), the only option to add a new icon set is to replace an existing one. I do not like the Crystal set, so I will just copy "images_flat.zip" over "share/config/images_crystal.zip" in the LibreOffice instalation path. Restart the LibreOffice and select the Crystal set in the preferences.

If you want to try the new flat icon set quickly, download images_flat.zip that is built from "stencils.svg" as of 2013-06-04 (updated).