The big challenge with the Roku3 is the one you've encountered: no dedicated audio output. There are very few HDMI switches that will actually split out the audio and video signals (most just pass the HDMI signal through). With no HDMI input on the TV or surround processor, your options are limited. The Roku 1, Roku 2, and Roku LT all offer analog audio outputs that would solve your problem - one simple fix could be to exchange the Roku 3 for one of those, at which point you could send analog audio to the 990 and use an HDMI-to-DVI cable to switch the digital video through the 990 to your TV (assuming it has a DVI input) or rely solely on analog audio and composite video until you upgrade the TV and processor.

The industry as a whole has pushed things heavily toward HDMI for a number of years now. This sort of problem will recur unless you upgrade at least the TV (and possibly the TV and processor) - new Blu-ray players have been phasing out analog video outputs, for example, as have some of the latest game consoles. You can find workarounds if you aren't ready to part with the old TV and processor, but they will take some careful planning when purchasing new sources.