Grim Fangio is ok. Broken sword is in a very similar vein to monkey island in the game play sense. However, there are 3 games that are alot more similar than the ones already mentioned by everyone else.
Firstly there is the Day of the Tentical. These guys also made monkey island, and not only is it funny, but is one of the hardest point and click games i have ever played. It is extremly clever.
Secondaly there is Sam and Max hit the road. Also made by lucas arts i think. Where day of the tenticle is more focused towards diffuculty, this game is absolutly hillarious, ive never laughed so hard at a game, and its a puzzle one to boot that is quite tough in places.
Finally there is an absolute classic called Benieth the Steel Sky. this was actually made before the original monkey island games, but has the same blend of comedy and puzzles.
The first two are a must for any Monkey island fan!!!