[Sprint Samsung Galaxy S III] topherk:
First off, you're going to need to have a custom recovery, if you don't already have one (most people do after they originally flash their phone). There are many guides to do this, just follow one of qbking77's great tutorials. I'd recommend using TWRP, but the latest CWM works well also.

Now that you have a custom recovery, you can try the following and it might get you out of the bootloop:

In the custom recovery, clear (wipe) both cache and dalvik cache. Then reboot the phone (reboot menu -> system).

If the above doesn't work to get you through the bootloop, try the instructions below:

Now that you have a custom recovery, you need to get the files onto your phone. Since you cannot turn your phone on to download it or connect through USB, you'll have to do it through recovery. Since I use TWRP, I'll walk you through the menu's for that (CWM should be similar).

Connect USB cable to computer -> Mount Menu -> Mount USB Storage.

Now your computer should see a "removable disk" with the contents of your SD Card on it. Now you can copy the ROM you downloaded over to the SD Card.

Now that the file is actually on the phone, you can flash the ROM zip in recovery. Whenever you do this, it's best to do what is called a "full wipe" in the recovery. You'll lose all your apps, settings, etc but it's much better than having a paperweight. In TWRP, what I always do is the following:

1) Nandroid backup: I normally name it something along the lines of "2013-03-14 - ROM NAME - ROM VERSION." This way, when I look through my backups, I can easily see what I was running (what I'm restoring when I want to restore).

2) Wipe Menu - Factory Reset. This will wipe all custom data off of the phone. For me, this is a must before flashing any new ROM. Other people will "dirty flash" Roms on top of each other. For me, that's a recipe for disaster, as any leftover settings/code will inevitably interfere with the new ROM and will mess things up.

3) Wipe Menu - Dalvik Cache and Cache: If I recall correctly, the Factory Reset does both of these, but it doesn't hurt to do it again.

4) Install: This is where you find the ROM zip file you downloaded and install it. Should take less than a minute, but some larger ROMs could take a little longer (not 10 minutes, though).

3) Wipe Menu - Dalvik Cache and Cache: Again, not technically required, but I tend to wipe the cache and dalvik cache very often.

3) Reboot Menu - System: Pretty self-explanatory, but it'll reboot your phone and you can set it up all over again.

In the future, it's best to have Titanium Backup to back up apps and data (For me it's worth the cost to upgrade to Pro). I have mine set up to backup all data every night at 3 am, with a maximum of 5 backups per app. This gives me almost a week of backups that I can restore for every app I have installed. The only thing to keep in mind with TiB is that you can backup/restore system apps that won't work with a new ROM. For example, if I backed up the Samsung Connection Optimizer on the stock rooted ROM and tried to restore it to LiquidSmooth (a 4.2.2 AOSP ROM), it wouldn't work. In fact, it could mess things up on my new ROM. So you might want to pick and choose system apps (RED items) carefully when you're restoring. Next time you want to try out a new ROM (or if you need to restore again because of a bootloop), you can just restore your apps/settings through TiB and not go through 2 hours of setting up all your settings again (for me I like that I can restore all the wifi access spots settings... saves me from explaining why I need my friend's WEP key for the fifth time this month).

Anyways, this post has turned into a novel. Please let me know if you have any more questions (and CNexus or anyone else can correct anything that's wrong)