I want to switch from my AT&T pay as you go cell phone plan to a
prepaid cellphone plan of another company.  Are there any companies
from which I can buy prepaid airtime without having to buy a new cell
phone?

I was just travelling to the states and wanted to buy something
similar. T-Mobile sells just a PAYG sim card for around $50. You get
some credit with that and then you just keep topping it up as you go.

The answer is.... MAYBE.
AT&T, Cingular, T-Mobile, Suncom, and a few other regional carriers
all use SIM cards in their phones. (some have been using SIMs for only
the past year or so) Also, every US carrier puts subsidy lock codes on
the phones they sell so that other SIMs cannot be used in the phones.
IF your phone uses a SIM card, and IF your phone is or can be
unlocked; then you can simply buy a prepaid SIM card from another
service provider and use it in your phone. Ask your local T-Mobile
dealer. They tend to deal with more world phones and international
users.

Note: Verizon, Sprint, Nextel, and Alltel will not be compatible