Grossi asks "... What do you think about instead?"
Hmmm ... I've never really thought about that 
But actually that is the answer! You don't have to be 'thinking' all the time!
When I wake up, I become aware of what sort of day it is outside.
Once I'm up I'll do some yoga before anything else. I enjoy the feeling of my body stretching out, and the peacefullness of the moves. No 'thinking' has been involved so far. As far as is possible, I focus on what I am doing right here, right now.
I've realised in recent times that I naturally live "mindfully" and have done so since way before it became a buzz-word.
For example, I really do notice everything around me, so whether I'm walking, driving or travelling on a bus or train, I will be appreciating the colour of the sky; the tiny buds on the trees in Spring or the varying shades of Autumn leaves. I'll see the squirrel that darts across the road or the urban fox sunning itself in a garden. I'll appreciate the wind in my hair or the rain on my skin. There is no space for worrying 
I actually find it very difficult to worry about anything these days! I very much feel that "Let it be" is all I need to feel. Life experience has taught me that things always work out. Maybe not in the way you thought you wanted, but they work out, and if necessary you take a turn and travel down a different path.
I have lived through times when I did not know how I was going to survive, and yes, back then I worried what would happen.
In those situations, I sat down and wrote out the worst case scenario, and speculated about how I would cope with those circumstances. The answer is that there is always a way, and once you realise that, and realise that you could actually survive - albeit in a less than ideal environment - the fear goes away. Maybe it is being reduced to the bare minimum for survival that makes you appreciate what is really important, makes you realise how little you really need, and makes you appreciative of what you have.
I found hitting rock-bottom strangely comforting. I realised I could cope and could still enjoy life, and maybe THAT is what made me a bone fide non-worrier 
Another important thing to realise is that there is no point in worrying about other people or their problems, because you cannot live their lives. You cannot fix their problems. It is negative and damaging to worry on behalf of others.
I try pointing out to those I know who are worriers that no-one but you has control over your thoughts! They are your thoughts. You have complete control over them! You choose to worry, so why not choose to think positively instead? Every time you find yourself worrying, make yourself turn the 'worry' around - find a positive opposite, or at very least a less-negative 'worry'. It might take practise, and writing your thoughts down will often help.
A good technique (I found) if your worrying is at a point that it makes you feel miserable or depressed, is to 'think a slightly less negative thought' - just one a day - but do it every day. Again, write it down. You might feel "this is hopeless, there's no way out, I feel helpless" A slightly less-bad place to be would be feeling angry (strange as it sounds) because saying "Why the @*! has this happened, it's so unfair" will in turn spur you on to action. If you can then think on the next day "OK, it happened, there's nothing I can do to change it" then perhaps the day after you can think "I can't change what happened, but I can make the most of what lies ahead, whatever it is"
Within a matter of days you can work yourself to a genuinely positive take on whatever the problem was! I found that over time this became automatic, and as I've already said, I find it very difficult to worry any more!
Finally, back to the 'mindfullness' It is the buzz-word these days because it does work for many people. If you are focussing on what you are doing right now, and being fully aware of the world around you, you simply do not have the space in your mind to worry
I didn't intend this to be a lecture
but I do feel very strongly about the physical damage that constant worry and stress causes the body, and it doesn't have to be that way. It really is all in the mind! So ... don't worry - be happy 