Is the kilogram still consistent with the old definition? My old understanding was that a gram is the amount of mass in a cubic centimeter of water. 

But then I recently learned that the second was redefined to be some 9 billion cycles of a Caesium atom. Then from this the definition of a meter changed from 1 ten millionth the distance between the equator and north pole, to $1/299 792 458$ the amount of distance that light travels in a second. From this I would have assumed that now a hundredth of that meter would redefine the gram, which would then be multiplied by $1000$ to get the kilogram. But I read that instead the kilogram is some reference mass kept in a safe somewhere.

Are these definitions nevertheless consistent with each other or are we just doing away with the linkage between the centimeter and the gram via the water cube?
The kilogram is likely to be redifined in the near future as a function of the number of atoms of a sphere of ultrapure $\ce{^{28}Si}$, as per the answer to this  Chemistry SE question. 
That means that the standard will be the Avogadro number, and the mass will be a derived unit. 

Then, any stored mass anywhere will just be an historical curiosity. 
This does not mean that it has not been important. 
We need exact measurements as Lavoisier taught to all future chemists.
At that time it was the best that was available. 
Now we might go to a still better standard.