I took Krav Maga for about six months before I started to get a persistent, nagging pain on the back of my left heel. I took Motrin (Vitamin M), but it still dogged me pretty consistently. I then took some time off from Krav to let whatever it was heal, but now that I'm back in it, I can feel the beginnings of it returning.

Has anyone ever gotten anything like this? Short of "Get your dumb ass to the doctor," do you have any advice?

well i dont know specificly but i do know what your talking about, my dad used to tackle me to the ground and squeeze between the bones above the heel, it hurt like hell and is immobilising, if you use enough pressure

Go To Your Doctor!!!!

That's what they are for.

Sounds like a bone spur to me and that ain't gonna get better.

Also try changing your sneakers.

Please expand on this, Vinnie! I was thinking today of the possibility that my shoes might have something to do with it. Do you have experience with this?

I'm assuming you will need to do alot of movement/bouncing, whatever.  Sneaker lose their "bounce after awhile, which means it's your foot taking all of the punishment.  Look at the bottom of your sneakers and see whether the wear is on the inside or outside of the sneaker.  After a while you will actually be standing like this:

\ / or / \ < your feet (exagerated of course)  looking at them either from the front or the back if you get what I mean

depending on how much wear is on your sneaker, so of course your joints will suffer. (Ankle joint/achiles tendon)

People who run and experience knee and/or lower back pain will inevitably find out it's the sneaker causing the problem.

Has the pain traveled to the sole of your foot yet?

Had something that sounds the same.
After about 6 weeks of pain. My wife dragged me to the foot doc.   It was called Plantar Fascitis (spelling)  I had an inflamation in the heal around the tendon.  Now I wear orthotics.  And I don't have the trouble any longer.  They will give you really nasty cortisone shots if you let it get too bad.  And if that doesen't work then you go under the knife.  I was lucky.  None of the above.
Good luck.  No one will laugh if you go see the doc.

If it's a "dull ache", like the kind you'd feel when having shin splints during running, chances are it's probably just an inflamed muscle/tendon.  In your case, describing where it's occuring, it's probably your tendon.

As far as techniques, I don't know what Krav Maga is all about.  Are you snapping your foot too hard when you deliver a kick to the head/midsection?  Are you doing any particular form that puts too much pressure on the back of the heel?  I can't tell you what to do (short of...you know what) to get it fixed, but the cause of the problem, which can be used for further prevention of the injury, can be taken care of.  

Chances are you're probably giving your kicks all you got, because I've never heard of any severe injury to the bottom of the achilles caused by repetitive kicking.  However, the next time it starts to ache, or if by chance you feel a sharp pain, then stop whatever it is you've been doing for the past 1-2 minutes....it's most likely that activity that's causing the problem.  This is "duh, no sh*t" advice, I know, but it will prevent you from furthering whatever injury you have until you see a foot doctor.