2009 Hyundai Accent misfires

2009 Accent - 1.6L
Getting P0300
What I have done - New plugs, new coil packs, swapped injectors to see if
misfire followed, new coil wires, tested cam position sensor and crank
position sensor, cleaned MAF sensor, did compression test.
Here's the weird - The original reading from the OBD was P0300-P0304 and
occasionally P0302.  Then started getting all three consistently, random and
2 and 4.  Finally gave up and took it to the dealership.  They first said it was
the coils.  I explained that I had already eliminated that possibility.  Then they
said they were communicating with Hyundai to diagnose the problem.  Then
they said it was a blown head gasket.  With no other symptoms indicating
head gasket issues I assumed they were just guessing. (Compression was
even on all 4 cylinders.)  Picked up the car and hooked up the OBD - now I
get P0300-P0301-P0303 ... consistently!  The misfire has moved from 2&4 to
1&3 and the dealer says they did nothing to the car.

What the heck could cause that?