Quick-and-dirty harmony question.

It seems to sound pretty much OK to have a harmony line shadowing the melody a fifth above - so if the lead is playing Three Blind Mice as B A G, the harmony line would run F E D.

But! If your original melody was in C, a fifth above that effectively puts you in G - so you've got the choice of either making the harmony follow the melody exactly (and keeping the interval fixed at seven semitones) by playing F# E D, or making the harmony fit the key of the melody (and avoiding possible dischords) by flattening the F#.

(Something similar would happen if you went a fourth above, only that way you'd be dealing with one more flat rather than one more sharp.)

But which is better? Is there a standard approach to this? If not, which way would people generally feel happier with?