Cut bread dollars (circles) that are about the size of the tomatoes being used. You can use a cutter or a glass. You can do a day or two ahead of time. Chop dill weed finely, reserving several sprigs for garnish. Blend the chopped dill with the mayo.
Closer to serving time (no more than two hours), spread the mayo mixture on the bread dollars (circles).
Slice the tomatoes thinly, and dab with paper towel to remove some of the moisture. Place one tomato on each circle.
Peel cucumber and slice into paper-thin slices. Place one or two cucumbers atop each tomato dollar. Garnish with springs of dill and serve chilled.