Cut lemons in half lengthwise and slice thinly, as if for garnish.
Mash them with the sugar and salt in a large, deep bowl or crock with an old-fashioned potato masher or heavy wooden spoon until the slices give up their juice, the sugar is dissolved and the juice is syrupy, about 5 minutes.
Pour about half the lemon slices into a sieve over a bowl or pot, pressing the solids to release as much liquid as possible.
Discard the solids and transfer the liquid to a serving pitcher.
Repeat the process with the remaining lemons.
Stir in the water until blended.
Add ice cubes or chill, stirring just before serving over ice.