The Nikon D600 has generally been free of complaints such as the AF point focus issues that plagued some D800s. However, one complaint I had heard about sometimes is oil spots on the sensor. Sometimes there's a little too much lubricant on the shutter. This is an issue that has affected the Nikon D7000 as well.
Oil spots look different from dust in a couple of ways: first they are visible at wider apertures like f/5.6 or f/8, unlike dust which is more likely to be seen at f/16 or f/22. Second, oil spots have varying brightness.
I took shots yesterday, and found what indeed look like oil spots on mine. :( See the top part of the shot above. I'm going to request Nikon to fix it. Hopefully they won't tie up the camera too long. Alternatively, I might ask for an exchange at Bel-Air Camera, where I bought mine (that's one benefit of buying from your local camera store). I'll update this post with the results of my request to Nikon or Bel-Air.
10/15/12 UPDATE: Picked up the D600 from Bel-Air Camera today. Their repair guys said they found no oil on the sensor, just dust. They said they were able to clean it with just a blower. I was surprised that the spots were just dust given all the characteristics I saw, and they said you can't really tell unless you look at the sensor with a loupe.
I took some test shots at f/8 through f/22. The spots that I saw were gone (there were a few specks that were visible at f/22 if I maximized contrast). See the test shots below. So it appears I didn't have any problems with my D600 after all. Meanwhile, one of the staff showed me how to use a sensor cleaning kit, which I will try on the D70 and I'll post about it then.
10/22/12 UPDATE: LensRentals' Roger Cicala has a related post http://www.petapixel.com/2012/10/22/the-nikon-d600-has-sensor-dust-issues/
Nikon D600 Resource Page (see under "Dust Spots")
Firefly DSC-2000 Review