Result of the experiment: It's possible, but the colour-switching is too fiddly for this to be fun. :D
The first variant is the technique described by Tinna Þórudóttir Þorvaldar, which involves crocheting each row from the right to the left and breaking the yarn at the end, without ever turning the piece. Each single crochet (US terms) is worked only into the back loop of the stitch in the previous row, and the double crochets making the pattern in the front loop of the previous-but-second row.
The second variant is based on instructions from Clearly Helena. This is turned and crocheted back and forth, switching colours every two rows. Wherever you want the colour of the next row to replace the colour of the current row, you crochet chains instead of single crochet, and then it's double crochets across those chains.
In contrast to the instructions given there, I did two stitches for each square of the pattern, because that way the result was closer to squares. Just one stitch per pattern square would have it stretched out vertically.
The first has the drawback that you have a lot of ends you have to weave in or otherwise take care of.
The second has the drawback that it's pretty easy to accidentally continue with single crochet where you should do chains. I think I crocheted each row twice on average. That's why I didn't continue up to violet, but stopped earlier.
That might be less of a problem with a simpler pattern and not having to pay attention to colour changes, too.
The first one is a little thinner and softer, but I'm glad I found and tried out the second one, so I have it available as an option.