I think the opposite of worry is trust.
Ponder on that for a fleeting second.
We worry when we don't know what the result is, but suspect it is going to be bad. When we know that whatever we are doing is taken care of, we don't worry. Because its just going to happen.
So to eliminate worry from your life, you have to introduce more trusted systems.
This is what David Allen is talking about in his popular "Getting Things DOne" organisational system. All the stuff about 43 folders and and obsessive compulsive dymo labling - all the productivity porn associated with GTD - is "How to build a trusted system". But the key secret behind GTD is that if you can get it to work, then everything is taken care of - you have no more worries, and you have more time to do things rather than work out what it is you have to do (and cope with all those things you've missed doing). The productivity porn isn't important - there are probably other ways to acheive a trusted system - what matters is that the system is trusted.
Now, as it happens, I've never made GTD work for me, because I can't fit the "how to" into my way of working... but getting a trusted system that I can rely on - that has to be a goal. I'm going to need to manipulate some of the ideas of GTD into something that fits my particular approach to life.
Trust comes from reliability and understanding. Reliability implies predictability - the system needs to give you the same results every time, and they need to be the results you want. Understanding means that you know why the system works, and you know what its limits are so you don't try to do something with it that it can't cope with.
Now, not all trusted systems are about automatability. It isn't all about lists and notebooks.
Some come from knowing your capabilities under fire ("if I do have to write this paper in one night, I can. I've pulled all-nighters before and I know I'm free the night before the deadline"), some come from having a safety net in place ("I know that if I lose my job, I have three months of savings to tide me over until I find another one - and Carl is always begging me to go and work for him") and some come from distancing yourself from identifying with things ("I lived without one of those for twenty years, I can live without this one if it happens to break")
So if you worry, its because you're missing some element of trust somewhere in your life. And that is because whatever you have in place either doesn't behave reliably where it counts, or because you don't understand the limits it works under.
Incidentally, this is as true with relationship worries (is my partner reliable, inso far as my needs are concerned? Do I undersatnd the needs and stresses that my partner is experiencing at the moment, that could limit her reliability?) as it is with worrying if you have enough milk in the fridge for breakfast (does my shopping system reliablymake sure I always buy milk before I run out? How does my system cope with unusual events like when I am sick? Do I have a safety net box of UHT to cope with the unexpected?)