I'm tempted to say non-Protestants pray to Saints, but that might be confusing, and misleading. Instead I'll make a short list: Catholic Churches, Greek and Russian Orthodox Churches, Coptic and Ethiopian Churches, "Churches of the East," Polish Catholic Church, and Old Catholic Church, as well as parts of the Anglican communion and some Lutheran denominations. There may be more, but I can't think of them right now.
Here are my past thoughts on this Saint intercession (as a Catholic):
We Catholics see the Church as a body in a very real sense: living things are wholes which have interdependent parts (you can't separate my heart nor brain from the rest of me without destroying me). The parts work intristically together. The principle that causes these parts to become one is called the soul, or spirit. The Holy Spirit is the principle who causes the Church to be one. The Spirit comes from Christ's intercession, which he then gives to us as he becomes one with the Church. Just as the Church is truly then one whole, with Christ as the head, whose parts work together for and with all the other parts, the Saints in Heaven are a part of the Church which helps the part of the Church on earth by interceding for us more Grace.
The Saints and baptized, by virtue of their rebirth ("born again in the water and the Spirit"), have the Holy Spirit dwelling within them ("we are temples of the Holy Spirit"). Grace to a Catholic is ultimately the Holy Spirit becoming "one" with us. Grace is therefore a share in the life of the Holy Spirit, and thus the very life of the Trinity itself ("the Lord and giver of life"). We live eternally because we live in the Holy Spirit. Original sin is essentially a lack of the Holy Spirit in a soul, which is why those who have Original sin can't get to Heaven, as the eternal Water who flows up to eternal life doesn't live within them. In other words, the Son took on human nature so that the Divine nature can "transfer" the Holy Spirit and Grace back those with human nature after the original sin. St. Athanasius says that "God became Man so that men could become gods." We become by Grace and the Spirit what God is by Nature.
The difference is that of an instrument and a member of a body. In the old Testament, the Patriarchs were instruments of God. Joshua was His sword, Moses His measure, and Isaiah His trumpet. In the New Testament, we are no longer just instruments of God's will, we are actual members of His body. St. Paul is His writing hand, St. Augustine His brain cell, and St. George His strong arm. For us to be His Body again, he took on a human one for himself.
In the West, we call this infusion of the Spirit/ infused Grace divinization. In the East, this is called theosis. So all the baptized, having the same Spirit within them, are one body, and as one completes all purgation of the connection with sin, and thus coming in perfect union with the Spirit, which finishes in the fire of purgatory after death (St. Catherine of Genoa says that purgatory is the purifying fire of the Holy Spirit within cleansing us: in a sense we are already in purgatory), the Saint realizes the full communion with all the other Baptized, as we all have the same Spirit within us. And the Saint prays to God out of Love for us, interceding for us. Since the Saint is a part of Christ's Body, the intercession is still through Christ. Thus, we pray to the Saint.
Because God now works within us instead of just through us, when we know through faith, we know with Divine Knowledge, and when we love with the supernatural virtue of Charity, we love with the Divine Love. Charity is the overflowing love of God via the Holy Spirit flowing from us to our neighbor.
The ancient Jews thought that by worshiping false gods, the gods came to possess their souls: God prepared the Torah and the Israelites by cleaning them of false gods, so that they might then become temples, clean from the filth of demons, for the Holy Spirit. We sin, and we allow the spirits of lust and the like to defile us, which exiles the Holy Spirit, since God will not dwell in such a divide and disgusting place. Sin (specifically mortal sin) is then literally the casting out the Holy Spirit from our souls, as if He were a demon
You might say the pure, satisfying Water of the Spirit (see the discourse of Women at the well in the Gospel of John) flows ultimately from the spring of the Father through the river of the Son, also can follow through the ocean of the Saints before he reaches us (this analogy is from St. John of Damascus). Water from the Spring doesn't come to the ocean unless it comes through the river though, and so the Spirit doesn't come from the Father except by the intercession of the Son (see the Gospel of John and Romans). However, the Saints receive in Heaven an overflow of Grace through Christ, which they can also give to us. So, when we are asking for the Saint's help, the Grace/Holy Spirit, which came from the intercession of the Son, flows to the Saint, and the Saint, overflowing with the eternal life (again, Woman at the Well), intercedes for us, and the Holy Spirit flows from the Saint to us as intercession. The Grace COMES from the intercession of the Son with the Father ultimately though.
By the Saint interceding for us, God sends his Grace through the Saint to us. Mary does this par excellence, as she is perfect, with the Spirit perfectly in her.
I think that the reason Protestants don't understand the Saint Intercession of the Apostolic Churches is in part because they reject infused Grace for imputed Grace.
Another reason, which I increasingly think is the main reason (IMHO), is that Protestants see prayer as worship per se. It is not. Prayer can be worship, but it can also not be. Liturgy/ Mass is worship in itself, though, and the basis for all worship. Because Protestants reject the Sacraments, and thus the Liturgy, they don't understand what real worship is anymore, nor prayer, or at least worship and prayer as an Apostolic Christian understands it.