+ 1It can be very deceptive where exactly the sound is coming from. I think I know the sound you're hearing seadragon. I've had it since day one if it's cold, and I thought it might be coming from below me when I started the truck but I checked once with the hood open and it was the power steering pump pully. After a few seconds it quiets down, I'd have to guess after the power steering fluid stops being a jelly and becomes somewhat liquid again.
I had the same noise on several cold starts recently. The first few times I heard it I was convinced that it was coming from under the middle of the truck. After opening the hood and leaning over the engine while my wife started it, it was very obvious that it was coming from the belts/pulleys on the left side behind the wiper fluid bottle. This makes sense because this is the first time in 2 years that the Tundra has been started in temps ranging from 10-25 F. It normally never gets that cold down here. The noise doesn't happen on a warm start or on a warmer day and the duration/pitch varies based on how cold it is so I am certain it is the belts. With the hood closed, the sound is bounced around a bit making it hard to tell where it is really coming from. I am going to squirt the belts with some dressing and wish for warmer weather to return.