There exists a heuristic that recommends checking from the diagonals (using a queen) in order to get a perpetual. This, I expect, is a rule that is meant for the worse player in queen and pawn endgames. My question is, why does checking from diagonals ensures, or at least raises the probability, for a draw?
As you said, this is a very general guideline, but looking back it has served me well in practical play.
Note that by "checking from the diagonals", the pattern I have seen is that the queen maintains a stable circuit on a diagonal but the checks themselves are usually horizontal or vertical.
I think it's because the checks from a diagonal allow the queen to keep the king in a box with only minimal pawn support (a single nearby pawn for either side will usually do it). Checks from a file or rank each allow the king to run across the board in one direction unless there's a pawn directly blocking his path.
I first heard this from chess24.com's GM Jan Gustaffson during one of his commentary sessions. The explanation he gave was that it is harder for the opponent to block diagonal checks with the queen. In queen and pawn endgames it can be much harder to continue with checks once the opponent's queen has managed to come back and help.
This personally makes no sense to me. Gustafsson might have been referring to a specific situation with other pieces involved, but if you experiment with a lone queen versus the king on an open board, you'll see for yourself that it makes no difference. Perhaps in a situation such as with the white king on b1 and his pawn on b2, and the black queen checking from d1, then after the white king went to a2, the black queen could continue checking diagonally from a4 to d1 and back to a4 consecutively, with the white king rotating between a2 and b1, until the position had been repeated 3 times, but that's a specific situation.