It's really weird that he would check this flop to you but realistically I'm not sure you could ever check back on the flop in this situation. Once he has check called the flop and checks to you again on the turn (given that there are no draws) then I think you could be suspicious enough to check the turn back because he looks like he's repping top set or AA or something.
If he then checked the river to you I'm not sure there would be a huge amount of value in betting and you could maybe check it down? Maybe this would be terribly weak play though? Would appreciate some thoughts on this....
If he bet the river then the decision to call or fold would depend on the sizing probably.
Basically - the fact he check called the flop and checked turn to you should make you v. cautious on this particular flop I think.