Mash cream cheese, mayonnaise, and ranch dressing mix together in a bowl with a fork until smooth; chill for at least 1 hour to blend flavors.
About 10 minutes before serving, remove cream cheese mixture from refrigerator and let stand to soften.
Spread about 1 tablespoon of the cream cheese mixture evenly onto each slice of cocktail rye bread, top each with a cucumber slice, and sprinkle with dill.