I figured it out! I'm thinking that everyone who tries to upgrade to Windows 8 from Windows XP and who uses the HDMI port on an ATI card for their surround sound (all 10 of us, in other words) is going to run into this same problem. A number of people building their Windows 8 machines from scratch probably will too. So, in case it benefits anyone, I figured that I'd share what I did:
I searched the folder where Catalyst was installed, and I found the HD audio driver there. The path to it is C:\AMD\AMD_Catalyst_12.6_Legacy_Win8\Packages\Drivers\WDM\HDMICE\W764a\AtihdW76.sys
On close inspection, I noticed that the date of this driver is older than the generic HD Audio Driver that ships with Windows 8. This probably explains why it didn't get installed automatically when I installed Catalyst. However, by going into Device Manager, and right-clicking the HD Audio Device and choosing Update Driver, and then giving it the path to the driver file, I was able to get it to take.
People upgrading from Windows 7/Vista won't run into this problem, because it lets you keep your existing drivers when you do that, but anybody installing Windows 8 on a machine with an older ATI card from XP or from scratch will probably find that they can't get surround sound over HDMI, even after installing Catalyst.
So, if that describes you, now you know what do to!