OK, it's happening now. The tray icon isn't rotating. I opened up the Avast UI and went to the Real Time Shields section. All the tabs there say "Protection Status: Running", and the File System Shield tab shows a couple of little yellow bars moving right to left along a red line at the bottom of a box labeled Shield Traffic.
The percentage of CPU time used by AvastSvc.exe (according to Task Manager) is in the low 90's, with AvastUI.exe using around 6-7%. After I closed Avast UI, the percentage for AvastSvc sat mostly at 99%, sometimes dipping as low as 95. All the other processes showed zero, with iexplore.exe and jqs.exe occasionally flashing a 2% for a second, then back to zero. The only applications I have running are Task Manager and a couple of Internet Explorer windows with no visible traffic on them (just static displays).