There are many possible ways of using the Psalter. When doing the Midnight Office in the morning, instead of saying the 17th Kathisma (Psalm 118) every day as appointed, insert one of the daily kathismata instead. That way you'd get through the whole things once every three weeks. The kathisma goes between Psalm 50 and the Creed, so if you're using a different set of morning prayers, this would be the natural place to read it.
Or, since each kathisma is divided into three stanzas, you could read 1/3 in the morning, 1/3 at noon, 1/3 in the evening.
Sun 2, Mon 4, Tues 7, Wed 10, Thurs 13, Fri 19, Sat 16.
Sun 3, Mon 5, Tues 8, Wed 11, Thurs 14, Fri 20, Sat 17.
Mon 6, Tues 9, Wed 12, Thurs 15, Fri 18, Sat 1.
Doing them in order might be more natural, but this way you'd maintain some link with the Church's liturgical cycle.