Ice Crusher
It might be fine to shake a cocktail with "household" sized ice cubes, but when you serve a cocktail "on the rocks", it is best to use something smaller. Either get several ice cube trays that make smaller cubes, or use some form of ice crusher to break the cubes up into smaller chunks.

Tall drinks such as a Mint Julep or a Tom Collins really need a large scoop of finely crushed ice.