Slashfilm has a wonderful article with a couple reports from the set of Cloverfield/1-18-08, which shot on Monday in Central Park. 
(Minor Spoiler Warning)
According to nooneimportant77, the first sequence involved (minor spoiler warning) a very bloody Rob and an unidentified girl (presumably Lily, played by actress Jessica Lucas) huddled under an over pass in Central Park. They exchange some lines, before Rob starts saying “look at me look at me”. Cue dust and bricks which fall on them while they scream. Then the production crew hoisted a big black tarp over one side of the over pass, which our source thinks was used to simulate more rubble falling.
Sounds like our friends from the trailer may play a bigger part than some people have speculated. And after watching some of the video, my thought is that the black tarp was used to block out the light. It's possible that the tarp could be used to simulate the monster(s) walking by. I'm sure time will tell.
Another interesting tidbit is that they were shooting with professional film or HD cameras, and not a consumer/prosumer grade camera.
The videos are now on YouTube. Thanks again to nooneimportant77.
Check out the full set report for more details, photos, and some video.
