From Nookipedia, the Animal Crossing wiki
The Rain Theme is a song that replaces all hourly themes in Animal Crossing (GCN) when it is raining in the player's town.
In Animal Crossing: Wild World and later, they no longer have a "Rain Theme" anymore, since the later games will always use the hourly music, regardless of weather condition, in a similar way to when it snows or is Cherry Festival day in Animal Crossing. In Animal Crossing: Wild World, it uses hourly music, but includes the tinkles and chimes as soon as it starts to rain (since the game uses sequences, banks and wave archives for music). In Animal Crossing: City Folk and later, they also use the hourly music, but they use streams (such as BRSTM on Wii, and BCSTM on 3DS, which is a raw audio format). The hourly music in Animal Crossing: City Folk and Animal Crossing: New Leaf have separate streams for each weather effect, one for clear weather, one for rainy weather and one for snowy weather.