The shield arrived and I directly assembled it. The data pins needed to be soldered on but this was really easy. It is a genius little thing because it comes with a built-in clock. This will make the data synchronization so much easier. Another plus are the numerous soldering holes on top.
OK, to get it running properly took a while. I forgot that some of the pins are occupied by the SD card writer. So when I connected some of the sensor pins by accident to these, I got really weird error which caused some headache. But after a couple of evenings with a lot of cursing, I think it is running properly now.
This is how the arduino with shield on top and connected to the sensors looked like:
The setup is so much easier now without the Pi. It will log data constantly as soon as it gets power.