When do you fold your non-nuts set? (LO8)

[TMTTR]

Hero in MP with A [img]/images/graemlins/club.gif[/img] K [img]/images/graemlins/spade.gif[/img] K [img]/images/graemlins/diamond.gif[/img] 3 [img]/images/graemlins/diamond.gif[/img] at a typical online low stakes game (3/6 or 2/4). Players are generally too loose but half of them have some concept of how to play post flop.

One limper, Hero raises, Button calls, SB folds, BB calls, limper calls.

Flop (approx 8SB): K [img]/images/graemlins/heart.gif[/img] 9 [img]/images/graemlins/heart.gif[/img] 4 [img]/images/graemlins/heart.gif[/img]

Checked to hero, hero bets, button raises, [either BB or limper calls], hero calls.

Do you now check/call, check/fold if unimproved? How does the plan change if it is not the button raise but instead BB or limper who check-raises? How does the plan change if you are heads up after the flop?

I find myself calling the river too much here (justifying my action because of big pot, etc.).