Heaney's main message in "Blackberry-Picking" is, "nothing's permanent, and we never get used to it," and that's what's important to remember. But Jesus Christ also plays a big role in this poem, complicating that message a little. If you think about it, it's the Christian belief to be all right with the death of everything on earth because there is Heaven to look forward to. From the Eucharist to the crucifixion, Jesus really dominates the symbolism. But keep in mind that although religion and religious imagery are big in this poem, that doesn't mean the poem is really "about" that. We wouldn't call this a Christian poem. It's much more universal than that. The Christianity just seems to be there because it was a big part of the speaker's youth.
The poem is about punishment issued by the Christian God for the speaker's sins of lust and greed.