Rav Hirsch's commentary on Tehillim (3:1) actually seems to say that one cannot make teshuvah without Tehillim.
Rebbe Nachman of Breslov also teaches that reciting Tehillim is an important practice for inspiring someone to truly do teshuvah. Here's a quote from his teachings on the matter:
Everybody wants to revere God's Name but not everyone is able to repent. Sometimes a person feels no arousal whatever. Even one who is aroused to repent may not reach his unique gate of Teshuvah, and even if he does, it could be that the gate is closed. This is why not everyone attains repentance.
But through reciting Psalms, even one who feels no arousal can be inspired to repent. The Psalms can take him to his unique gate and open it up, thereby bringing him to Teshuvah.
For this reason King David called himself "the man who raised the yoke... the sweet singer of Israel " ( II Samuel 23:1) . Our sages explain that David called himself "the man who raised the yoke" because he elevated the yoke of repentance through his own Teshuvah. David was a great Tzaddik and should not have sinned, but God caused him to sin in order to teach everyone the way of Teshuvah. King David was the prime exemplar of Teshuvah and his pathway is set forth in the Psalms, which he wrote with such a spirit of holiness that everyone can find himself in them and thereby return to God.
Likutey Moharan II, 73