One of the perhaps lesser-known works of Thomas Aquinas is his Catena Aurea, the Golden Chain. This is a commentary on the Four Gospels, one verse at a time; Aquinas gives the verse, and after (nearly) every verse intersperses commentary on it—notably from Augustine, Jerome, and John Chrysostom; though other theologians, Church Fathers, and Doctors of the Church are included if they have anything to say on a particular verse.
Aquinas himself comments on Matt. 6:34:
Having forbid anxiety for the things of the day, He now forbids anxiety for future things, such a fruitless care as proceeds from the fault of men, in these words, Be not you anxious about the morrow.
In addition to Aquinas' own gloss, he quotes a similar idea of Jerome:
Sufficient for us is the thought of time present; let us leave to God the future which is uncertain. And this is that He says, The morrow shall be anxious for itself; that is, it shall bring its own anxiety with it. For sufficient for the day is the evil thereof. By evil He means here not that which is contrary to virtue, but toil, and affliction, and the hardships of life.
In other words, Jerome says that Christ means something like, "The future is going to have enough problems; worry about them as they arise, not now."
John Chrysostom says something similar:
Has not every day a burden enough of its own, in its own cares? why then do you add to them by laying on those that belong to another day?
Aquinas also quotes Augustine of Hippo, Hilary of Poitiers, and an author known as Pseudo-Chrysostom. This unknown 5th-century author wrote a commentary called Opus Imperfectum In Matthaeum (Unfinished Work On [The Gospel Of] Matthew). These three apply a more metaphorical approach to the verse.
Augustine, for example, comments
Tomorrow is said only of time where future succeeds to past. When then we work any good work, we think not of earthly but of heavenly things. The morrow shall be anxious for itself, that is, Take food and the like, when you ought to take it, that is when necessity begins to call for it. For sufficient for the day is its own evil, that is, it is enough that necessity shall compel to take these things.
In other words, he says, Jesus is warning us not to focus on the things of this world, the things we'll need from day to day. Instead we should focus on the things of Heaven. We should do what is necessary to survive in this world, but not allow it a greater place in our lives than that.
Pseudo-Chrysostom makes a similar comment:
Why then should you be anxious about those things, time property of which you must part with? Sufficient for the day is its own evil, as much as to say, The toil you undergo for necessaries is enough, do not toil for things superfluous.
Finally, Hilary applies this comment to the necessity of repentance and the possibility of salvation:
This is further comprehended under the full meaning of the Divine words. We are commanded not to be careful about the future, because sufficient for our life is the evil of the days wherein we live, that is to say, the sins, that all our thought and pains be occupied in cleansing this away. And if our care be slack, yet will the future be careful for itself, in that there is held out to us a harvest of eternal love to be provided by God.
"Sufficient unto the day is the evil thereof", in this conception, means something like "You have enough to do each day in focusing on repenting for and avoiding the sins of the day."
That appears to be a reasonably complete survey of major views of Matthew 6:34.