Many people still wait until well into June before planting, especially frost sensitive plants. Well, grandma had a good reason to say this: in June, the soil should definitely be warm enough, and there should be no frost. Grandma had no accurate weather forecasts like we do, and no access to historical first and last frost dates.
Now we are all about extending our growing season which is around 130 days from around Victoria Day until mid September. Plant early in the season, and protect your plants from frost in the Fall. Plant most greens in early May, once the soils dries enough so it is not soggy. Many plants do better in early spring rather than during the heat of the summer (spinach won’t even germinate if the soil is too warm). Plastic tunnels over your beds will warm the soil and protect early planted frost sensitive plants), watching the weather forecasts for frost and protecting your plants when frost threatens also works.
So Grandma was right, but today we can get a jump on spring with careful planning.