depends on whether the tile runs under the cabinets.

don't even dream of travertine until you have the floor joists checked to see if they are stiff enough for stone flooring - most are not. Yours is even more suspect since you already have cracked tile. Might not be stiff enough for ceramic either. You are going to have to remedy any subfloor problems that are likely underneath that tile now.

Proceed carefully.

I agree. It could be one of many things causing your floor to crack, and every one of them has to do with movement getting to the tile. Whether it means there was no thinset under the cement board, or maybe ther WAS no cement board, and they used too weak a thinset over the plywood, or (and this is the biggie) the floor isn't structurally sound enough to accept tile (and for stone, it has to be doubly as strong as for tile or any other flooring).

You need to find out what the problem is first before you decide to put stone down. Almost guaranteed, even once you find the problem, as antss said, the joisting will have to be beefed up, as well as maybe even the subfloor.