What you're considering has a name: app-o-rama.

More here: http://online.wsj.com/article/SB118255545661145353.html
and here: http://www.app-o-rama.com/

Inquiries will be logged immediately, but most card issuers don't pull all three credit bureaus, so your inquiries will be spread out. You also won't be turned down for "too many new accounts." Some banks/card issuers (Barclays/Juniper) are very sensitive to new accounts/inquiries, and they have been known to close accounts after they have been opened (called Adverse Action or "AA").

So as long as you're not planning to buy a house or a car anytime soon, good luck, and avoid any card issued by Barclays/Juniper