I finally delved into the owner's manual of my F150 an realized that yes, you can disable the annoying seatbelt chime without altering hardware or adversely affecting other warnings/indicators.  It's right in the manual, which is the last place most people will probably look (at least, it was for me).  This is said to work on other Ford vehicles, as well.

Temporary, One-time Disable

After you have started your vehicle, buckle and then un-buckle your safety belt.

Permanent Disable/Enable

This is a multi-step process, so read it over before you get started.  Before disabling/enabling BeltMinder, make sure that:

• The parking brake is set
• The vehicle is in park (automatic) or neutral (manual)
• The ignition is off
• All doors are closed
• The driver's safety belt is un-buckled
• The parking and headlights are off

Then, follow these steps in order.

1. Turn the ignition switch to on, but don't start the vehicle
2. Wait for the safety belt warning light to go off (about 1 minute)
3. Within 55 seconds of the light going off, buckle and un-buckle the safety belt 9 times, ending with the belt un-buckled
4. Wait for the airbag light to flash on, then back off
5. Buckle and unbuckle the seat belt one more time
6. If done properly, you will see the safety belt warning light 4 times per seconds for 3 seconds indicating that the enable/disable is complete

Now, just because I told you how to do it doesn't mean you should do it.  I am also not encouraging anyone to not wear (or wear, for that matter) a seatbelt.  This information is provided as-is for anyone who is simply wishes to disable the BeltMinder feature in the way that the owner's manual suggests.  Your vehicle may vary, so for complete instructions refer to your owners manual.  For my 2004 F150, this information can be found on page 119.