Besides the obvious theological reasons, I don't do yoga because I find it insulting to the real practitioners of yoga: those individuals devoted not just to the bending, stretching, and strengthening that the practice offers, but those also who are also devoted to the spirituality of yoga. Those who are trying to be yogis.
Those who only do yoga for the physical benefits are not really doing yoga. Without the spiritual aspect of the yoga the person isn't actually doing yoga. They are just bending and stretching and breathing deeply, all things that can be done within the Catholic tradition.
All in all, those who wish not to participate in the spiritual aspects of yoga turn it into an attempt to unspiritualize something that has its roots in the spiritual. It is like trying to separate the rosary from Mary and the Mass from Christ.
Lastly, for a person to tell me that they only pray the rosary because it makes them calm or because it is healthful to their body makes me want to slap the person who just told me that as they use a practice of my faith as a mere physical tool that leads the person not closer to Christ but turns them inward.
So I don't do yoga because I choose to be somewhat sensitive to another person's spiritual belief and practice and chose not to trivialize their faith by making it into a mere physical exercise.
The Jesus Prayer
So maybe you want to clear you mind and meditate and breath deeply in the Christian Tradition. Try the Jesus Prayer. It is a prayer that is near to 2000 years old, and it requires the person to clear the mind and focus the mind on Christ and God's mercy. The prayer goes as follows: Lord Jesus, Son of the Living God, have mercy on me a sinner.
Traditionally the person repeats this prayer over and over again. A deep breath is taken and the first half of the prayer is prayed (Lord Jesus, Son of the Living God) and the end of the first half of the prayer the second half is prayer during the exhalation (have mercy on me a sinner). A rope or rosary can be used not so much as to count the prayers but to help concentrate.