Some other critter would evolve to take their niche. Not sure how long that would take though, but eventually there would be other insects which will annoy you.

There are many species of cockroaches, most of which aren’t pests to humanity. Just living their little bug lives out in the wilderness munching stuff. And getting eaten by insectivores in large quantities.

Humans wouldn’t notice much at first, but things would seem off and get worse over time. Disappearing animals - birds, reptiles, fish. Angry frog owners pacing back and forth. Orkin men begging in the streets.

Probably a significant quantity of decomposing matter not being cleaned away and becoming more and more noticeable.

Nothing bad

You’ll notice at least one person that is happier about them ceasing to exist. (That person is me, I hate roaches)

My pet tarantula would probably be a bit sad.