The way I pray when I am in the midst of daily life is not quite the same as the way I pray early in the morning when I can be alone and quiet and undisturbed by life's business. As you point out, it is not easy for most of us to mentally "walk out" and "concentrate on prayer without thinking of anything else".
For me, the best solution has been to stop trying to do the impossible. Instead of trying to make the busy-ness of life go away, I invite God to come into the busy-ness with me. I can still go on thinking of whatever it is I am doing, but the conscious awareness of God's presence in the moment turns the activity itself into a kind of prayer.
To give some nuts-and-bolts examples of this method as it works for me:
I am waiting on customers, and there is a line, it's a hot day, I'm tired, and I'm hassled. I am getting a bit short and brusque with the people, pushing them through like widgets on an assembly-line, with an expression and manner that says, "You are bothering me, go away." I am certainly not mindful of God at the moment. Then, I take a deep breath, and I think, "He's here. He's standing in line, right in front of me. He's behind the counter with me, helping me work. He's here." I turn to the next person in line and really pay attention to that person, really extend "good vibes", with a manner and expression that say, "It's you! I'm so happy to see you here! What can I do for you? How can I help make your day better?" It's not just a matter of wearing a fake pasted-on smile. It's a whole attitude change. God is there, with me, and it is as if the air were suddenly charged with electricity, as if I had just seen my Love walk into the store and come towards me. I go on working, but the work has become a kind of prayer.
Another example: I am doing some kind of tiresome uncomfortable chore -- raking leaves, let us say, or shovelling the drive on a sleety frigid morning, or washing a sink full of greasy nasty pots and pans. I am feeling down and self-pitying and whiney about the whole thing. "How come I always get stuck with these jobs? My feet hurt, I'm sick of this, I wish I was sitting in my armchair with a cup of tea." Then I invite God into the picture. He is my honored guest, and this meal is being cooked for him, these pots and pans are being washed for him. This driveway is being prepared for him to walk on, this yard is being cleaned to celebrate his visit. Don't we humans go "all out" to spiff up the place when the president comes to town? When the public eye is on us? When the boss is coming to dinner? This is even bigger and better -- God himself is here! This is now a glowing honor I am engaged in, not a drudging chore.
Of course, I try to do the other, more meditative kind of praying, too. Usually early in the morning, when the world is still and dim and the day's noise has not yet broken my concentration. But when life is busy, I can't just "shut it off" and meditate. The "pray-as-you-go" method works for me. Maybe it won't work for everybody, but it can't hurt to try it and see.