A common cause of groans and creaks (that appear only when there is weight or force on the pedals) is insuffiently tightened crank bolts, so if it were my bike giving trouble, that's where I would start, torque wrench in hand. There is considerable disagreement about whether to grease crank bolt threads. You can ask for opinions here; I grease mine because if I don't, they creak, and I've never had one spontaneously unwind even though I torque them to only the lowest rating, 30N/m (you need to check what's specified for your particular cranks, or torque them to 35N/m which is a common lower limit for alloy cranks and not above the upper limit for any alloy cranks I know).
After that, from the easiest and most obvious causes:
I imagine you have long since checked that the chain isn't rubbing somewhere!
Have you fitted particularly fat tyres? If so, check by dusting the closed ends of the forks with talcum powder or rubbing on wax (one of your wife's decorative candles will do) and riding the bike, then checking what rubbed off from the fork to the tyre.
Does the width of your chain match your chainring? You must be very careful here. KMC makes chains with tiny variations in thickness (which many of us discover when we buy the wrong width of quick link), and the Thorn chainring (if that's what you have) is particularly thick presumably for reasons longevity.
Do you have enough slack in the chain? You should be able with the bike at standstill to depress the middle of the free run of the chain at the top by at least 10mm. A Rohloff flourishes on a much slacker chain than would be acceptable on another drive system.
Have you checked that the chain is running straight? Rohloff advises that the chainline should be within 1mm of straight. I mention this because, while you don't say which KMC chain you have, the single speed KMC chains are deliberately made very stiff from side to side as a factor in their longevity.
You may have a small amount of latitude from side to side in fitting the bottom bracket. After you've measured how far the chain line is off, inspect the bottom bracket; you want to know the length of the axle and whether there are any spacers already fitted, or space for you to fit spacers to put your chainwheel in a more advantageous position vis a vis the chainline to the sprocket.
If there is not enough adjustment on the bottom bracket, Rohloff used to give away with every gearbox a set of spacers which can be used with extra-long chain bolts for spacing the chainwheel; if you didn't get the chainring spacers with your bike, or have lost them in the years intervening, you can buy a packet of spacers from Rohloff dealers like SJS.
The last possibility I want to mention is that somehow you misfitted the bottom bracket. If that is the case, and you have corrected the problem while sorting out the chainline, you'll never know what precisely caused the problem, the chain or the bottom bracket. Small price to pay to have the problem gone.
If the problem isn't solved by this stage, and if you haven't taken off the bottom bracket, now is the time to take it off, and inspect it, and also the thread in the BB shell, thoroughly. When you refit it, use a torque wrench (common automobile type) especially on the nuts that hold in the pedal cranks.
If your bike still emits disturbing noises, it's gremlins, known where I live as leprechauns. Have you toured in Ireland, Wales or the Continental Celtic countries recently?
