The idea of determining the effect of a particular type of poem is interesting but probably also open to a rather broad interpretation. For me, the primary effect of free verse poetry is that of narration. In other words, these kinds of poems often sound as much like storytelling of various kinds as they do poetry.
Robert Frost's "Home Burial" is a good example. This free verse poem is in our textbook and we usually read it together, but I tried an experiment one year. I simply read it aloud to them, without trying to either "play up" or diminish the poetry of the work. When I finished reading, I asked for their impressions. Almost without hesitation, several students remarked that it "almost sounded like a poem." So there it is. I know this poem is more of a narrative (story) than some other kinds of free verse, but another Frost work--"Mending Wall"-- sounds as much like a story as a poem, despite a few more poetic devices.
Other works, such as Whitman's Leaves of Grass, are less obvious but also tell a story. (Whitman's work is more of a stream-of-consciousness narrative, another kind of storytelling).
You may get a few other perspectives here. I'll be as interested as you to read them!