Do Tehillim follow a specific schedule corresponding to the days of the week?

Yes. There is a way the chapters can be divided evenly so that the entire sefer can be said over the course of a week. The order is as follows:

  • Day 1 (Sunday): Tehillim (Psalms) 1 – 29
  • Day 2 (Monday): Tehillim (Psalms) 30 – 50
  • Day 3 (Tuesday): Tehillim (Psalms) 51 – 72
  • Day 4 (Wednesday): Tehillim (Psalms) 73 – 89
  • Day 5 (Thursday): Tehillim (Psalms) 90 – 106
  • Day 6 (Friday): Tehillim (Psalms) 107 – 119
  • Day 7 (Shabbos): Tehillim (Psalms) 120 – 150

Source: Sefer Tzeidah LaDerech, 4th maamar, klal chamishi (this source has a slightly different breakdown of which chapters end and start each day, but it is THE original source. Artscroll Tehillim, and many other sources have the breakdown as outlined above)