Are there any local charity organizations or religious type groups such as the Salvation Army accepting of basic household item donations?

Based on what your profile says, you're in Tokyo so this should work well.

You can probably get rid of a lot of stuff through Tokyo's Garage Sale section. Basic Sayonara sale stuff.

I'll re-iterate what nadiatims said. Hardoff and Off House sell tons used kitchen stuff as well. You can sell/give practically anything you can think of to Hardoff/OffHouse.

Do you have a have a valid license? If you need to you can rent a Kei truck for a day/few hours and probably move any large items. But yes, I believe Hardoff and them might pick up some items.

So I realize that there's other stuff I'm not certain of either -- last time I was here the people at the BOE helped out a lot.  Is there a nice checklist somewhere of things you need to do before you leave?  I'm not sure how to handle bill payment, or whether I might get some of the 敷金 back (and if so how I would get it from the US).

I also wonder what I do after I get rid of all my furniture...I guess I might have to stay in a hotel for one day.

Call the local city office. As a resident, you can get a discount to dump your junk at the local landfill/incinerator. If transportation is an issue, you can probably pay a "dump it all" truck to come and take everything away. It might be 10,000 a load, maybe more. Ask for a number from the city office. Also, ask your neighbors/co-workers and such. The locals know more than anyone.

On checklist.

Make sure all billable things are canceled. You may be able to arrange to pay the last bill right then or receive a conbini payment. This way you can close bank accounts if you want (no reason to).
Cancel health insurance.
Make sure you don't have any taxes that need to be paid to the city.

Assuming the place hasn't been damaged too badly, you should be able to get some/most of the deposit back. It can be tricky, but there are normal wear-n-tear things that can't be taken from the deposit. I know tack holes in the wall for instance can't be counted.

I recommend googling on it. Tenant laws are very strong in Japan and the laws dictate the kinds of things that the landlord can claim from the deposit. Its the typical language barrier thing and the foreigner being ignorant of it all though, which is why most people believe the deposit never comes back. There may be questions on Okwave or Yahoo Chiebukoru with pertinent links that can help you figure out where your rights lie.