I honestly think that there are very few truly non-liturgical churches, in a broad sense. Certainly some of the Quaker churches are truly non-liturgical in all sense of the category.
In reality even non-liturgical churches have a set liturgy, even if they make it up. You know the one, upbeat song, upbeat song, solemn song, introduction by pastor or worship leader, upbeat song, sermon, upbeat song, dismissal. It's the same liturgy every week, the same rhythm.
That said, when we talk of "liturgical churches" we are usually talking about more Traditional churches with a liturgical tradition that may (but does not have to) also follow the common lectionary. The non-liturgical idea is almost exclusive to Protestant traditions. In general the Anglican communion, Lutheran traditions, Catholic Church (Eastern and Western), Orthodox (Eastern, Oriental), some Reformed, and other "high" churches are liturgical. However, there are also "low" church Anglicans and others that would still qualify.