If you can get LEDs which use a true/pure/ripple free DC supply then you won't get the flicker.
Depending on how much they are willing to spend it would be worth giving a company like photon star a call and asking if they can come up with a suitable driver for their eco600 down lights (which are the dogs danglies of LED d/l in my opinion) or similar. You might find the non-dim able driver with a high quality fitting like the Eco 600 doesn't strobe anyway (but I doubt it as they are all switch mode)
Failing that you could construct 12V linear DC supplies with LM7812s, a transformer and a couple of capacitors to supply MR16 LED lamps which shouldn't need huge transformers to be used.
I don't normally advocate the use of LED retrofit lamps like these, but in this case it might be the best solution.
It is possible to make linear constant current drivers too, but they won't have the efficiency of a switchmode jobby.