June is the month of the Sacred Heart: a celebration of God’s love for us.
Music: O, Sacred Heart, Sweet Heart of Jesus, To Jesus Heart, all burning, Oh the Love of My Lord is the Essence, My God Loves Me, ; God so Loved the World (from the Stainer crucifixion)

Quite separately, I also think of it as a good time to buy roses.  I think it maybe used to be the most seasonal month or something. 

Other things in season: asparagus, maybe; broad beans, should you like them (I find it hard to believe that anybody does); lettuce, e.g. little gem, or the floppy types; crab, apparently; sardines, fresh, grilled; strawberries; cherries (foreign, probably); gooseberries.