I have dedicated lines, but why is this happening?

It may be time to look at the A/C unit as the source of the popping. Areas I would look at include making sure all the electrical connections are tight and corrosion free. Obviously do this with A/C breaker off. Don't overlook the exterior disconnect box as those contacts may be showing signs of corrosion/oxiditation. Also you may want to see if the A/C is low on refrigerant as this can cause increased load during start up.

As far as Peterd's suggestions, I've never been a fan of floating grounds (ie a ground bar that is not tied to system ground) and they are specifically prohibited by the US National Electric Code due to shock hazard potential. For what it's worth ground rods work better in moist soil and the recommendation is to locate them outside at least 1 foot beyond the spill line of the roof so as to take advantage of natural rainfall.

I also agree that perhaps the AC is the culprit. Perhaps it's now drawing more current at startup.

But I would seriously doubt grounding is the issue.

-IMO

Dedicated does not mean isolated. Your dedicated lines are tied together with everything else at the power panel so it is very easy for noise generated by something like the AC unit to find it's way into your gear.

When an air conditioner cycles, there are relay contacts that are actuated. These relays commonly arc when they are actuating, whether opening or closing. The older they get, the more they arc. This arc throws wideband hash into the air and onto the power line. It is similar to a small arc-weld. This is very likely to be the culprit. You probably need better RFI/EMI protection/filtering in your system.

Two things you can check:

The central ac unit may have a problem such as a faulty compressor (overheats?), pull-in contacts, loose wiring on the motor, loose wiring in the circuit (wire nuts, j-boxes, etc), loose fan belt on the blower, or a bad low-voltage transformer. I would bet on the contacts that pull in the blower line voltage.

The work done by the electrician. Each circuit have its own neutral home-run? Sometimes they'll cheat by grabbing the nearest white wire. Big no-no.

All motors cause transients and surges in your house power system. The above can make it worse, in terms of hearing the noise in your gear. If everything checks out ok (including your gear), then you may look into getting a whole-house surge suppressor (TVSS) installed in your main panel. I also agree it ain't a grounding issue and that dedicated circuits are not the cure-all for power problems.