There is a Service Bulletin out for 2012 Civic HVAC noise.. titled.."Buzz or Rattle from the dash when the Heater is On"
The symptom is "a buzz or rattle is heard from the dash when the heater is on low to medium fan speed"
Honda states the possible cause as "The air mix door is warped" The fix is to replace the heater Sub-assembly.
That description sounds a little different from the sounds people have been describing in this thread but that is the only HVAC bulletin out for 2012 Civics, including the hybrid and natural gas models. If your sound is anything like "amirdieselx64's" then I don't think it can be fixed because I heard absolutely NOTHING in that 51 second clip. I was waiting and waiting and there was no sounds there??
If you go to the dealer and this sounds like your problem tell them to look at Service Bulletin 13-041.
Here are the part numbers for the Sub Assemblies listed in the bulletin:
PARTS INFORMATION
Heater Sub-Assembly (4-Door EX, LX, Japan-built):
P/N 79106-TR0-A02
Heater Sub-Assembly (Hybrid):
P/N 79106-TR2-A42
Heater Sub-Assembly (4-Door EX, EX-L, LX, Si, HF,
Natural Gas, U.S-built):
P/N 79106-TR3-A02
Heater Sub-Assembly (4-Door DX):
P/N 79106-TR3-Y01
Heater Sub-Assembly (2-Door EX, EX-L, LX, Si):
P/N 79106-TS8-A02
Heater Sub-Assembly (2-Door DX):
P/N 79106-TS8-Y01