I think you will mostly be alright, cards were accepted pretty much everywhere but having a handful of small notes and coins would be handy. In terms of Euros there may be the odd tourist orientated place that will take them, but for most small purchases it would be much more useful to have local currency.
If I remember correctly most of those WHS have free entry, so that removes one problem (Jelling, Tanum, Christiansfeld, Stevns Klint, Parcforce, I haven't been to Karlskrona or Oland). Grimeton had facilities to pay by card, I would imagine Roskilde is the same, but sometimes cathedrals can be a little less modern in their approach.
At Stevns Klint you will need some coins (40kr) to pay for parking (only coins were accepted) you will have to go a long distance away from the visitor centre and church to find free parking.
Credit Cards were fine for the Bridge Crossings, and actually save you some money if you are crossing the same bridge twice (probably not relevant in your case), though be prepared for it to be eye wateringly expensive (Storebælt: 34 and Øresund: 46).
Most shops, restaurants and petrol stations will be fine with credit cards, but there may be the odd one that doesn't have the facilities so good to have a few kronor in your pocket. If you are staying at a Bed and Breakfast at any stage it may be good to have the money in local currency as they aren't always equipped with a card machine/online payment.
Just as a quick note, if you are heading over Storebælt the t-list site of Trelleborg Fortress
is just nearby (another review I have been meaning to write) it is actually one of the most interesting and tangible Viking sites I've visited (much more interesting than Hovgarden or Jelling in my opinion). Also it is free to visit, apparently it is closed on Mondays, but I think you could probably just walk up to it, avoiding the visitor centre. It is marked up on the Motorway, but just keep your eyes peeled for the small sign once you get on the country roads.