The experienced defender has agreed a two-year deal with the Dongwall club.
Munro made more than 300 appearances for Inverness before he was released at the end of the season.
But Caley Jags gaffer Terry Butcher was happy to release the 30-year-old as he rebuilds the squad for an attack on the SPL top six next season.
"I am delighted to have signed Grant," said County manager Derek Adams.
"His experience on the pitch will be vital in the season ahead.
"He has been a consistent performer over many years and will add further quality to our squad."