Probably a significant quantity of decomposing matter not being cleaned away and becoming more and more noticeable.
Also they convert that to nitrogen so our soil could be less fertile if they weren’t around.
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.
The king is dead, long live the king!
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.
You’ll notice at least one person that is happier about them ceasing to exist. (That person is me, I hate roaches)
I will join you in the celebration of no roaches.
Lots of unemployed exterminators.
Nothing bad
think of all the jobs that would be eliminated (exterminated?) if there was no need for cockroach traps/spray!
My pet tarantula would probably be a bit sad.