If you are like me and you don't have a HDMI tuner that is capable of switching HDMI signals and extracting the digital audio signal for its own use, chances are good that you are sending video via HDMI and audio via optical or coax (optical being the better choice). There aren't a whole lot of solutions out there that allow you to switch between multiple HDMI inputs and multiple TosLlnk (optical) inputs hooked into a single HDMI and Toslink output set, but I was able to find one switcher that won'e totally break the bank: the Octava 4-port HDMI Switch with Optical Toslink Routing (that's a mouthful, so I'll simply refer to it by its model number, "HDS4").
As you can see from the picture above, the HDS4 does exactly what I would need it to do: it provides HDMI switching for video and Toslink switching for audio from multiple inputs to one output. Furthermore, the HDMI ports are backwards-compatible with DVI, meaning that you can plug a DVI-to-HDMI cable in from a DVI source and not have any sort of output problems. Here's an illustration from the Octava site that illustrates what an implementation might look like:
It is important to note that the HDS4 solution is remote-driven, not an auto-switching solution. In my case, I'm happy to look at remote-driven solutions because if a product is RF- or infrared-driven, then chances are good that the folks at Logitech have the product in their database of devices that work with my Harmony 890 (I checked, and the HDS4 is indeed in the Logitech product list).
Although a little more expensive than some HDMI switching solutions at $274, I have not been able to find other solutions that also include the optical audio switching, so this seems like the best solution for my particular needs in the short- and long-term (I currently would only need 2 of the HDMI and Toslink inputs) and the fact that it is remote-driven makes it extremely easy to integrate into my existing system with very little trouble.
Take a look at the link below and also note that Octava does package the switcher with cables, which, if 6ft HDMI cables will do it for you, are pretty good deals.