Just got the Roku HD-XR. Is it possible to use the Roku to play internet radio stations and our own music (for example, mp3 and iTunes files on our own external HD) without keeping the TV on, by using the Roku non-HDMI ports (we use the HDMI ports on the TV for the Roku and another device) to connect to external speakers and say using a laptop to control the music?
My experience is that it works, yes.
Just cable up the ROKU to your stereo's auxiliary input, in addition to cabling it up to your TV. You then use the TV to find what you want to listen to, and then turn on your stereo (and adjust it to the auxiliary input) and you can continue to listen via the stereo even if/when the TV is off. And as an extra bonus, the audio from such an arrangement is often of higher quality than you get via a TV (which is why I'll often use the stereo to get the sound from a movie, instead of getting the TV's sound directly).
For example, I have my ROKU's left-right (analog) audio stereo outputs cabled up to a stereo, and the HDMI cable going to the TV. Since I'm using different ROKU outputs on each device, I don't even need a Y splitter cable to make things work. However, even if you use the same audio outputs for both devices, you can still use Y splitter cables to hook both up at the same time.
NOTE: If you don't have a full stereo that has support for an auxiliary input (and some cheaper stereos don't these days), you can also make do with just a set of powered computer speakers (i.e. any computer speakers that you plug in, because they contain their own amplifier). With the powered computer speakers you do have to get a little creative with the audio cables used (because you will need to convert from the ROKU's left-right stereo audio output jacks into the other gender mini-stereo plug most computer speakers use), but it does work with the proper audio cables (I tested that setup for a family member, and verified it worked).
NOTE: I use the ROKU's left-right audio jacks for "audio only". However, if your stereo supports the ROKU's digital audio output jack, that would presumably be another option to consider.