Well, the question as stated :
What do gravitons interact with?
is naive enough . It assumes gravitons exist in the way photons and gluons and Z exist.
In such a framework, the answer is : they interact with everything for which a Feynman diagram can be imagined.
The other three forces' mediating particles (photons etc.) are absorbed by their appropriate charge-carrying particles, is not quite correct.
The force mediating quanta interact with all matter, even if not at first order, through higher order loop Feynman diagrams , is the correct formulation.
but I can't seem to find a clear answer that applies to the gravitational force in a quantified scenario.
You mean a quantized scenario.
In string theory, where the Feynman diagram formalism is appropriately modified for the extension to strings, a graviton will be a mediating particle on par with the other "forces" mediating interactions.
As was discussed in another question, the forces are an artificial carry over from the macroscopic classical world. What exist are interactions that are characterized by coupling constants which distinguish if the specific interaction is strong, gravitational, supersymmetric or...