How can I extrude only the black part from a QR-code?

You would need to create faces to be able to extrude them. You could trace the QR code with the Line and Rectangle tools to create the edges for the faces or you’ll need to find some way to get vector lines you can import. One option might be to create a PNG in which all the white areas are transparent and use TIG’s Image Trimmer extension (from the Sketchucation Plugin Store.) It may be able to handle it but I haven’t tried it to know for sure that it would work.

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If you have a “crisp” image in BMP format. you can use this plugin to import it into SketchUp. As @DaveR points out, you will need to extrude each appropriate region after deleting the ones you don’t want. More discussion can be found here:

Here’s an example model created from a 1658 x 1658 QR code:

qr.skp (211.3 KB)

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Convert to a vector graphic…(might have to scale the qr-code image way up to avoid the converter thinking it’s just very pixelated and turn them into curves)
http://vectormagic.com/home

then svg to stl…
http://svg2stl.com
One of the gallery items on the site shows someone has already done a qr-code…
http://svg2stl.com/convert/5790a822c90967df0aa96c3c

Import and play.

My result.