I never was able to figure out what was causing the sound. I changed the oil with Mobil 1 0W-40 and the sound pretty much went away. After about 4000 miles the sound has come back. I had an oil analysis done by Blackstone and it showed some bearing wear. Regardless, it's either the oil pump or something being lubricated by the oil. Overall my car has been making this sound off and on for more than 20000 miles so I'm not sure what to make of it.
I'm probably going to change the oil again soon and see if it makes the sound go away again (and run another oil analysis).