No matter what tradition you believe in, chanting helps prolong the exhalation of the breath, calm the mind, slow the heart, and after long periods, helps bring the meditator into an alternate state of consciousness. Whether this is due to the effect on breathing, the effect of one-pointed thinking, the effect of the repetition, the vibrations of the mantra itself, or (most likely) a combination of the four hasn't been proven, but chanting or reciting mantras does seem to be a highly effective form of meditation for people of many beliefs, traditions, and cultures. It might be just the technique for you.
You don't need to know any particular tune in order to chant. You can simply chant your mantra on one pitch that's comfortable for your voice.
Here are some possibilities for mantras if you don't already have one:
One is all, all is one
Let it be
Shalom ("peace" in Hebrew)
Om shanti shanti shanti ("All is peace, peace, peace" in Sanskrit)
Om (The primal sacred syllable)
Kyrie Eleison ("Lord have mercy" in Greek)