After further investigation, I can at least partly answer my own question:
My mistake was that I was trying to impose the microprocessor's timing (using a timer interrupt) onto the AD7176, whereas the AD7176 was using its own master clock to control the channel sequence and conversions. Therefore the result was random.
I was initially not interested in the chip's filtering capabilities, but I should probably have read that section of the data sheet, too, because that section contains the essential clue. The FILTCON registers allow you to control the output data rate.
I have now started implementing a different way of accessing the AD7176. I'm now using a GPIO interrupt that triggers on the falling edge of the MISO (alias DOUT/RDY) line. This interrupt then reads the data from the SPI. Still not working 100%, but that's a different issue.
Hope this helps for anyone who stumbles into the same problems.
cheers
Hendrik
P.S. maybe one could use the SYNC line to work with the microcontroller as the master clock, but I haven't tried this.