Paul, it is not "completely wrong". Its not a great way to do it if you want to achieve a low multiplex ratio, I grant you.
I was attempting to re-draw the OP's original circuit, but using my preferred "linear" arrangement.
Multiplexing does work with resistors on both anodes and cathodes, the OP already found this out by accident. The current has to pass through 2 resistors, so those resistors should be a lower value to maintain good brightness/current. But it would effectivrly restrict you to lighting one led at once, ie a 1:10 multiplex ratio to light all 10 leds, which might be bright enough, depending how bright the leds are. Not much good for 64 leds though, for example!
If the common anodes were connected directly to the Arduino outputs, more than one led in each group could be lit at once, reducing the multiplex ratio down to 1:4. But the series resistors would need to be higher to avoid overloading the Arduino output driving the anodes. So for good, low ratio multiplexing, transistors would be the way to go, giving a bright, evenly lit display.
I have read Tom's blogs before and found them interesting. His charlieplexing explanation is good, but I still find my linear idea easier to understand, personally.