The dimmers are spewing radio-frequency hash into the power line, which confuses the AFCI into a false trip. It makes sense that it's more reproducible on an intermediate dimmer setting, because that's when the dimmer chops the line voltage aggressively, not when it's fully on or fully off.
Bypass capacitors could help with this: 1 to 10 nF capacitors, X1/Y2 rated ceramic or film, minimum 250VAC working voltage rating. This diagrams shows three caps: between hot and neutral, hot and ground, and between neutral and ground.
The basic idea is that 60 Hz AC will not pass through these capacitors because the frequency is too low, but radio frequencies pass through easily. So the capacitors provide a short circuit shunt path for high frequencies, reducing how much of it spews back upstream from the dimmer.
This looks simple, and it is, but it has to be done properly. Capacitors have long, uninsulated terminals, creating opportunities for short circuits if you're not careful.
Instead of these ad-hoc capacitors, you can install an EMI filter upstream from the light dimmer. EMI filters have a more sophisticated circuit which includes inductors.
One more thing: the Lutron dimmer company sells a filtering component that they call a lamp debuzzing coil.
Best Answer
Most microwave ovens will generate arcs and sparks inside their cavity if they are run with an insufficient load, or no load whatsoever. The high-frequency content from these arcs is being coupled (likely capacitively) through the HV transformer section of your microwave onto the AC line, where the AFCI can then see it and proceed to freak out, thinking the arcing is actually an AC mains arc, not a microwave cavity arc.
Suggestion: throw a coffee mug of water in with the slice of pizza. This does the same thing as Keshlam's sacrificial bread slice -- i.e. provides more load to absorb the microwaves instead of letting them bounce around the cavity until they arc, just without having to waste a slice of bread.