My approach to davvening involves elements of meditation and acting technique. When starting out a session of davvening, the main problem is that my mind is anywhere but focused, except for a few seconds at a time. Psukei Dezimra gives me time to correct that. The shortness of Minchah and Maariv make that difficult. I simply take my time. My favorite technique for this is dwelling on each word at once, and as I say it, allow its implications and its meaning to be felt. For when davvening beyechidut, I get to take the time to really feel each word fill my body and be expressed with the full force of what I can feel of its meaning.
When I say "feel its meaning," I mean that in the sense that certain colors make us feel a certain way, or certain people's tones of voice or qualities of body language lend a kind of feeling or flavor to them. This is the kind of meaning that can't be articulated verbally, except by poetry, but only experienced on an emotional and physical, and eventually spiritual, level.
When I'm really doing this well, and I'm really open to the liturgy, and really being carried by it and really hearing/feeling it, extraneous thoughts tend not to be too much of a problem because I'm wrapped up in an experience. To really use this technique to the fullest, let each sensation of each word move your body in whatever way it's moved. The full flavor and power of each word will fill you. To best do this full, physical version of the technique, take a single pasuk, and repeat each word as many times as you need to in order to feel all the different nuances of the word before moving on to the next one.
That is an acting technique I learned from my teacher Marcia Taylor-Croft in college. She taught it to us in a studio class that focused on using archaic, heightened text (ie: Greek dramas, Marlowe, Shakespeare, De La Vega, etc); taking text that because of its dialect and its hyperbole, would be alien to our natural states of mind, and owning that language, and expressing ourselves through it.
Like I said, this technique is all well and good for when you have time, but most minyanim speed-davven, and for the shorter davvenings like Minchah, Maariv, or Musaf, you simply don't have that extra fifteen to twenty minutes to reach a meditative state through focused awareness. So I choose a brachah. The one that you halakhically must have kavannah for is Avot, so I take my sweet time with Avot.
R. Aryeh Kaplan z"l in his book "Jewish Meditation" recommends taking about seven seconds for each word in the Shma. I do it for Avot. Sometimes a simple, quietly observant meditation on the words, and sometimes a toned-down version of the acting technique that I described above.
Another thing that keeps me focused and opens me up to the spiritual power of prayer is directing the words to God. Talking to God, and using the words of the liturgy as my own words. An actor, when delivering a monologue, or acting out a phone conversation, needs to imagine the other person listening, and give his or her lines to that other person; say the lines, as your own words, to that other person. The difference for a praying Jew is the faith that the God you're directing the words of prayer to is real, not imagined. Again, experience the words, but instead of dwelling on the words' internal power and meaning, focus on who you're saying those words to. The power of the words becomes apparent in an incidental sort of way.
I've been reluctant to teach these techniques because I haven't mastered them. I often lose focus when I davven, and to be an effective and safe meditation teacher, I need to have mastered the art of meditation. Still, I feel like making the techniques known for readers to explore on their own terms is a safe enough thing to do.