I’m all for any spiritual practices (ancient, like this one, or innovative) that help us pay attention to God.  You might want to try this one!
This document on Praying the Hours, is based on Seven Sacred Pauses by Macrina Wiederkehr.  These are just some suggested Psalms for each pause.
The Night Watch (midnight)
A deep, even dark prayer of waiting and interceding.
“My eyes are awake before each watch of the night, that I may meditate on your promise.” (Ps.119:148)
Psalm 42
Psalm 63
Psalm 119:145-152
The Awakening Hour (dawn)
Remembering God’s goodness…thankful for a new day, a fresh slate.
“Satisfy us in the morning with Your steadfast love; so that we may rejoice and be glad all our days.” (Ps. 90:14)
Psalm 19
Psalm 95
Psalm 147
The Blessing Hour (midmorning)
The Blessing hour is about being aware of the Spirit’s abiding presence ability to make the work of our hands something sacred, marked by love.
“Let your loveliness shine on us, and bless the work we do, bless the work of our hands.” (Ps. 90:17)
Psalm 67
Psalm 84
Psalm 121
The Hour of Illumination (noon)
We honor the hour when Jesus embraced the cross.  We pray against darkness that consumes, steals and ruins.
“You are the salt of the earth…you are the light of the world.” (Mt. 5:13-14)
Psalm 24
Psalm 33
Psalm 34
The Wisdom Hour (mid afternoon)
This hour embraces the themes of surrender, forgiveness and wisdom.
“For to me, to live is Christ and to die is gain.” (Phil. 1:21)
Psalm 71
Psalm 90
Psalm 138
The Twilight Hour (early evening)
The themes are gratitude and serenity as the evening lamps are lit.
Psalm 34
Psalm 139
Psalm 145
The Great Silence (bedtime)
A gentle evaluation of the day focusing on awareness, teaching us to be healthy sinners, living in neigher denial of our sin or despair because of it.
“By day the Lord directs His love, at night His song is with me – a prayer to the God of my life.” (Ps. 42:8)
Psalm 23
Psalm 91
Psalm 134