What is Ujjayi?
When done properly, Ujjayi (translated as "victorious") breathing should be both energizing and relaxing. In the Yoga Sutras, Patanjali suggests that the breath should be both dirga (long) and suksma (smooth). The sound of Ujjayi is created by gently constricting the opening of the throat to create some resistance to the passage of air. Gently pulling the breath in on inhalation and gently pushing the breath out on exhalation against this resistance creates a well-modulated and soothing sound—something like the sound of ocean waves rolling in and out.
The root of your problem may be as simple as the effort you exert to perform Ujjayi. It is important to remember that the key to Ujjayi breathing is relaxation; the action of Ujjayi naturally lengthens the breath. Some small effort is required to produce a pleasing sound, but too much effort creates a grasping quality and a grating sound.
To practice the inhalation, focus on creating a soothing and pleasing sound that is unhurried and unforced. I suggest working on your Ujjayi breathing in a seated, relaxed cross-legged position. Imagine sipping the breath in through a straw. If the suction is too strong the straw collapses and great force is required to suck anything through it. Once Ujjayi breathing is mastered in a seated position, the challenge is to maintain the same quality of breathing throughout your asana practice.
Throughout your practice, try to maintain the length and smoothness of the breath as much as possible. Once you find a baseline Ujjayi breath in a pose that is not too strenuous (Downward Facing Dog for example), endeavor to maintain that quality of breath throughout the practice. Some asanas require great effort, and you may begin to strain in your breath.