Updated (14:17 p.m. EST): Checking back at the site a few minutes ago, the Obama White House now has a press briefings section with the Briefing Room portion of the site. You can find it here.
This morning I discovered that the Obama administration's WhiteHouse.gov is indeed transcribing and maintaining its daily press briefings featuring Press Secretary Robert Gibbs. Of course you pretty much have to know where to find them since there's no link to an archive section from either the main page or the "Briefing Room" page.
By entering "press briefing" in the search field, I got a hit for the first Gibbs press briefing on January 22, 2009. The link is as follows:
http://www.whitehouse.gov/the_press_office/Press_Briefing_1-22-09/
To access the other daily press briefings, simply go to http://www.whitehouse.gov/the_press_office/Press_Briefing_m-dd-yy/ substituting the numerical values for the month, day and year accordingly (e.g. /Press_Briefing_1-26-09/).
This writer has no idea when the briefings were actually published to the WhiteHouse.gov site, however, as each briefing is listed as published on midnight of the day of each briefing.