A Christian seeks a new mind, the mind of Christ. He cannot simply empty first, then fill. The mind rages against that, and will seek to occupy the mind again with something from it’s reservoir of memories or imaginations. The journey of a Christian, seeking to grow in prayer, is to seek to gradually replace the mind filled with this futile world, with the holy truths and hopes and concerns of Christ. The best sources to go to, to begin this, are Holy Scripture and the Catechism, and/or the trustworthy prayers and writings of the saints of the Church, and/or the doctrines and teachings of the Church – in effect, the Truths of God, as passed on to the Church.
Lectio Divina presents a way to meditate upon Holy Scripture in this way (at least, the reference I suggested does this. Some interpretations of Lectio Divina focus far too much, imho, on the self and not enough on divine revelation as it is in Scripture.).
After some period of time in this practice of Christian meditation, the person will, or may, find himself meditating upon smaller and smaller portions of Scripture, or simpler and simpler expressions of holy Truth. This is a good sign, and indicates growth in prayer, with increasing attention and devotion - mind and will - on the truths of God. If the person continues in this way, the time may come when the next phase of prayer comes upon the person: infused contemplation. Infused contemplation is not something that the person does, or can do. It is a gift, an act of God. Many “teachers” of prayer may try to “teach” “contemplation.” If they do, and if the student thinks he is “doing” infused contemplation, he probably is not. If a man can do it, it is not a gift. If a man can teach it, it is not a gift infused by God.
Some teachers can lead a person through vocal prayer, into Christian meditation, and on into greatly simplified meditation called “the prayer of simplicity.” This is close - in some senses - to the “mindlessness” that you may be speaking of. This prayer can also be called “acquired contemplation.” Again, this is NOT the same as “infused contemplation”, nor is it the “meditation” of Eastern religions. It is meditation - mental prayer - upon simple and beautiful truths of God.
The best teachers of prayer for Christians are Christians, following the great masters of prayer - John of the Cross, Teresa of Avila, and (maybe surprisingly) Thomas Aquinas. Nearly contemporary excellent teachers are Fr. R. Garrigou-Lagrange, OP and Fr. Jordan Aumann, OP. Their books are free and on-line, in many cases. If you want to learn about prayer, those are the men to read.
I hope this helps.