Most gamers probably aren't aware that a sequel to Chrono Trigger actually existed in the past. It's true, Chrono Cross isn't the first game to carry on the Chrono Trigger story.
Three years ago, a multiscenario digital novel game by the name Radical Dreamers was released for the Super Famicom add-on peripheral Sattelaview. The main story involved characters by the name of Serge and Kid - the same characters currently appearing in Chrono Cross - and Gill. In Radical Dreamers, the three attempted to steal the Frozen Flame from Yamaneko's manor. As the story progressed, there were a lot of indirect references to what happened to some of the characters from the original Chrono Trigger. The relationship between Kid and Yamaneko introduced in Chrono Cross is actually explained in the story of Radical Dreamers.
The big question now is how much of Radical Dreamers' story will be carried on into Chrono Cross. Sadly, since the game was only released on the Sattelaview, there's a good chance many of these tales will be long forgotten.