Thank you Golfwhenucan. I also have an old Airport Express (model no. A1264 - can only be read with lots of light and a magnifying glass!), which I use to get my laser printer onto my home network. I also tested using this Airport Express (optically connected to the DAC) instead of the AppleTV 3. Result: no noise! So I can confirm your finding.
Now you say that the new Airport Express suffers from the same problem as the AppleTV 3? I was about to buy one of those new Airport Express thingies. I guess I could always use the new one for the printer and just use the old one for music streaming. It is kind of silly though, since I already have the AppleTV close by my stereo and TV - and this is SUPPOSED to be able to handle this job! Unfortunately, my AppleTV is just a month or two over two years old - and where I live the warranty is two years. The thing is, I've not tried to stream music to it before, so I didn't notice the problem before it was too late, as I didn't own a DAC.
By the way, I believe my old Airport Express is second generation (model no. A1264), even though it's the kind that plugs directly into a wall socket. So I guess the current Airport Express (similar to AppleTV in shape) is third generation.
Finally, I would like to ask you an additional question about your Marantz HD-DAC1:
Mine is in for repair right now, because I experienced noise when using the front iPod/iPhone usb port. The noise could only be heard when using headphones or the variable out on the back panel. In other words whenever the volume knob was engaged simultaneously with the iPod port being used. I don't have an iPod, but I was using both an iPhone 5c and the latest iPad Retina. The noise level was constant irrespective of the setting of the volume knob (except the noise would almost disappear at close to maximum volume), increased when increasing the Amp Gain setting, and was sensitive to display changes on the iPhone/iPad (more noise when using the touch interface). So here is my question: Have you tried to connect an i-device to your HD-DAC1 via the front iPad port? - and have you experienced anything like what I've described?