As I understand your description:
- The GFCI protection device is in one location in the kitchen
- The circuit breaker in the main panel does NOT have GFCI protection built in (if it does then things are a bit more complicated having double-protection)
Before doing anything else, disconnect EVERYTHING from all of the affected kitchen outlets. If you can then reset the GFCI and the outlets function then you have a faulty appliance - plug each appliance back in until you find the problem. It is quite possible that there is an intermittent ground fault so that things could work for a while and then stop working. But in the end, if you have a faulty appliance then either fix it or throw it away.
Replacing the GFCI and troubleshooting other outlets should be done with the circuit breaker turned off for safety:
Bad wiring or outlet on "load" of GFCI outlet
All the other affected outlets should be connected to the load side of the GFCI outlet. To determine whether the problem is in the GFCI device or one of the other outlets, disconnect the wires from the load screws. If the GFCI can be reset and the outlet works then you know the problem is on the load side. If that is the case then you need to reconnect the load wires and check the wires to each outlet one at a time until you find the culprit. One thing to watch for is back-stab connections - if you find any then switch to using the screw terminals as they are more reliable. Loop connections are OK if installed properly - tightened well, correct amount of insulation removed, but pigtail connections (which require a little more work) are better.
If a new GFCI does not work - or if you feel like checking before installing a new GFCI: Check the wiring going in to the line side of the GFCI to see if it is live. The safest way to do this is with a non-contact tester. If the wiring is not live and the breaker in the main panel has not tripped then you have a wiring problem somewhere in between the breaker and the GFCI.
Bad GFCI device
The outlet with the GFCI device may simply be going bad. That can happen. It is fairly straightforward to replace one, and that is often the simplest way to diagnose the problem. If the problem goes away with a new GFCI outlet then either the old one was bad or the wiring into the old one (which should have been checked by the electrician) was bad.