In this spot, I generally check, bet, bet, and fold to any raises whatsoever. It's rare that I've actually ever had to check-call down because most people would try to trap with JJ here, for example. They would check flop, call or raise turn, or raise river if they tried to slowplay the flop & turn. But let's say he does not slowplay JJ here. We check-call two barrels and check-fold river. AK should not be 3-barreling for value against us, and it's super unlikely anyone tries to 3-barrel us when our range is face-up to TPGK/TPTK.
Without the Ks, it is okay to c-bet since he can have combo draws like KQss and KJss sometimes. Checking and c-betting without Ks are both okay.
People are not going to bluff at this board texture much, so don't worry. I used to be wary of checking such boards, thinking my opponents might try to bluff me. But people generally play more honestly in 3-bet pots, and they aren't going to try to rep the A if they don't have it. I remember this one deep-stacked reg on Zone 100NL once typed, "KK huh. I should shove lol." He gave up after a weak turn bet. Everytime I got barreled off in a 3-bet pot when I was the PFR, it was generally very strong.