tcrooks0904,
I have the exact same board (GA-HH87M-D3H) with an Intel i3-4130T, 4400 graphics, same latest driver, and I have the same problem (e.g. HDMI connected to TV via an Onkyo Receiver). So long as the TV and Receiver is on and set to the input the PC is connected to, I can get audio and video through HDMI. As soon as I switch to another source on the receiver, or turn off the TV/Receiver, the Intel Display Audio device disappears from the list of usable audio devices and doesn't come back until a full reboot. I've tried to set up everything to "5.1", even though my receiver is capable of 7.1, I ran into some odd errors when setting Windows to use 3.1 (I didn't have the surround speakers set up yet), but setting to 5.1 got through those errors and started to expose some others (e.g. Dolby error below).
I've opened a support case with Gigabyte, they've come back and asked to confirm the motherboard model and asked if the problem went away if the TV was direct connected (something I didn't try as that's not a useful setup for me, but I'll try and report back to aid troubleshooting).
I've also noticed a few other quirks within Media center, specifically trying to play a DVD through Windows Media Center (Windows 8.1 64-bit) gives me a display error (seems to be tied to anything that tries to use Dolby). Note that this worked fine in my old media center w/the same TV/Receiver and an ATI video card.