Well, after traveling through 14 countries in the last few years, I finally managed to lose my passport. That's fairly impressive for me; anyone who knows me knows that I lose phones, keys, wallets, and other objects other people typically keep up with on a fairly regular basis. Luckily, it's not the end of the world for US citizens living in Bangkok to get it replaced. Just don't plan on leaving the country anytime soon.
The American Embassy is located about a 15 minute walk from Phloen Chit BTS station. Exit the station from Exit 5, turn left, and keep walking. Note that before reaching the consulate-ish area where you actually get your passport and other documents, you will see "the embassy" on your right. The building you're looking for is on the same side of the street (your left).
You can probably find it by the immensely long line of Thais waiting to get documentation of whatever kind. I feel a bit bad about that; there is a "foreigner" line and a US citizen line. Just go up to the window (there typically isn't a line for citizens the few times I have visited), tell them you need a passport, go through the security rigmarole, head into the US citizen services building nearby (follow the signs), fill out the paperwork, and you're pretty much done. I was surprised how easy it was, actually.
The fee as of May 5, 2013, was $130 if you pay in USD; the exchange rate they're currently using is good if you have USD--31 baht to $1. Not bad! But, you can pay in baht if you like. It was something like 4,080 baht for the fee.
Don't worry if you don't have a copy of your passport; they seem to be ok with it. I had one (as it's always a good idea to do), but I saw other people who did not have a copy with them. Some kind of ID will obviously be helpful, but everyone in the office was quite nice and helpful, so I'm sure they will find some way around it if you lose everything.
After you pay and fill in the relevant paperwork, they will give you a receipt and tell you to come back in roughly two weeks to pick it up. It only took them a week and a half for me. I was brilliant enough to let my wallet with my passport copy and other IDs get stolen off of the beach in Phuket, so when I returned, I simply showed them the email they sent me saying to come in to pick up my passport, and I was golden.
The entire process, both applying for the passport and picking it up, didn't take more than an hour each time, security included. It's the first thing I've ever done with the US government that ran so smoothly, and I'm quite impressed. And now, I have a brand new passport with 10 years of validity! Let's hope I can keep up with this one...
Don't forget, though: you have to go to Bangkok Immigration and get a new entry stamp. That process SHOULD take ten minutes (although mine took all day due to them not having any record of me landing at the airport weeks before), and will require four things: a copy of the police report from your lost passport, a letter from the embassy saying you got your passport replaced, a copy of the new passport information page, and finally a form you pick up in the immigration office that summarizes all of this activity. The new entry stamp is free, unless you have overstayed your visa from your last passport.
If you have overstayed, as the officers were convinced I had (having no record of me being in the country), you must pay the typical 500 baht per day overstay -plus 1,900 baht for a 7-day extension visa. You don't have a choice. I finally gave up and ended up paying 4,000 baht in fees because it was no longer worth arguing with them, but hopefully that doesn't happen too often.
Still, for the passport itself, it's not a terrible process. Just hope you don't incur the wrath of immigration.