Most of the major banks offer online bill pay (Bank of America, Wells Fargo, Bank One, etc.). It's so convenient because the bank itself is making my payments and, of course, it is all done over a secure server via the Internet.
If your bank does not already offer this service, there are several sites on the Web more than happy to let your pay bills online through their sites such as CheckFree, PayMyBills.com, or Paytrust (which will even balance your checkbook, for a fee!).
Many of these sites even allow you to download your information to financial programs such as Quicken or Money so that you don't even have to enter your bill paying activity in your checkbook program.
Not surprisingly, the U.S. Post Office has missed me (& the many, many other people already doing online bill pay). If your bank doesn't offer bill pay and you are not sure you want to sign up with some site out there on the Web, you can pay your bills online through the USPS site itself!