I have the same issue, 2010 SXT purchased in November 2010. I was wondering if I was the only one and have had my vechicle at the Dealership twice, I actually recorded a sound clip of the noise so they could hear it. Just had my second visit back to the dealer today, after dropping the vehicle off last night. The first visit they replaced the power steering pump, resivor and fluid however the noise was still there the next morning. Returned the vehcile for a second visit and they said they checked the belts, pulleys and the tried three other new 2010 Journeys on the lot and they all made a similar noise. The Dealership said they had spoken to Chrysler and they are aware of the noise however there is no TSB so there is nothing further they can do. They did provide me a printout from Chrylser stating the noise is acceptable for 1 second at temperatures of 0 F (-18 C) and -20 F (-29 C) for 15 seconds. I talked to the mechanic and he was quite confident it was an problem with the pump design. I asked the Dealership if I could hear the noise from another new vehicle on the lot and it was a similar noise, but not quite as loud and shorter in length.
For me this isn't acceptable for a new vehicle, hopefully Chrysler comes up with a fix for this, it's embarrasing to start my vehicle first thing in the morning.