Who cleans their AK with a solid rod from the breach?  It seems some receivers are cut low enough in the back to allow a rod without bending, others are almost impossible.

Yes, I have always cleaned with a rod from the breech...I use Tipton CF rods.  Its really not an issue on the AK.

1) Put the rod in at the muzzle withOUT a jag, loop or brush screwed into it.
2) When the end of the rod appears at the receiver end of the barrel, attach your preferred tool end.
3) Pull the rod out, chamber to muzzle, and the brush, etc. along with it, it doing its cleaning task along the way. No rod bending can occur.
4) Remove the tool end, wipe off and reinsert the rod, repeat these steps as needed. (... or you slip the unit armorer a $20 to accept the rifle as-is)

BTW, you just learned how I clean a M1 Garand, M1 Carbine, M14/M1A or just about any lever action rifle, too. 

Ditto with my AK74.  You bend the rod slightly, but only slightly.

The cap on the cleaning kit is made for the rod to fit through and over the crown to protect
it during cleaning. The AK was made for battle conditions and designed thus. I will see if I can find my tutorial
on the cleaning kit.

Here is one to get you started. https://www.youtube.com/watch?v=2H3IlpQnlxg

Here is another and there is one more but if you see this yb vid you can find the
other. Even the cleaning kits were designed smartly. https://www.youtube.com/watch?v=Wh2VORpYRMA

Yep. Get an Otis kit - you won't be sorry. You see, we don't have to use battlefield-expedient measures when we clean our guns during peacetime - there's plenty of time to do it in the least-harmful way possible which is to clean from the breech on any gun.

... and I’m going to replace my old brass jags with a full set of these aftermarket caliber specific pull-through style cleaning jags from ProShot, for use with that Otis kit that I bought in the mid 90’s and wasn’t really impressed with. They’ll also work on a regular cleaning rod:

http://www.proshotproducts.com/Tactical-Line-Replacement-Brushes--Jags_c_110.html

I use a cleaning rod from the breech with no problem and for a quick cleaning a Boresnake does the job.

There's no reason to stick a cleaning rod down the muzzle.