(These instructions may seem long and complicated but don't let that daunt you from trying this. The actual method is pretty simple and straightforward. In my usual style, I am giving you tips and advice based on my experience with this method.)
(Method 1)
You need:
5 pounds Roma or large grape tomatoes, very ripe but not ones that are getting soft
Fine sea Salt
You will also need metal cake racks, vented pizza or cookie pan or cookie/pizza sheet
Instructions
Preheat oven to 200 degrees F. , or the lowest setting possible. Remove the oven racks except for the center one.
Trim and discard the stem ends of the tomatoes. Halve each tomato lengthwise. Arrange the tomatoes, cut side up, side by side and crosswise on cake racks set on the oven racks. Do not allow the tomatoes to touch one another.
If you are using a smaller type of tomato, you can dry them on a vented pizza pan (they look like they are made from wide mesh or just have holes poked in them) You can also use a regular cookie sheet or flat pan but you want to use one with very low sides, so the air flows around them. You will need to stir them periodically if you use a solid pan.
Sprinkle lightly with fine sea salt. Sometimes I sprinkle with an Italian herb or garlic salt mixture, to create a seasoned version.
Place in the oven and bake until the tomatoes are shriveled and feel dry, anywhere from 6 to 12 hours. Check the tomatoes from time to time. IMPORTANT: They should remain rather flexible, not at all brittle. (Smaller tomatoes will dry more quickly than larger ones. Remove each tomato from the oven as it is dried.) Once dried, remove the tomatoes from the oven and allow them to thoroughly cool.
Transfer the cooled tomatoes to zipper-lock bags. Properly dried tomatoes will last indefinitely. I always put them in the freezer to be on the safe side, however. You can process them in jars of oil for long term preservation, but I have never used that method so I won't describe it here.
To use them, I take the dried tomatoes from the freezer, thaw them and pat them dry with a paper towel. Then I pack them in a small jar and cover with olive oil. I also pack in a couple of peeled garlic cloves and let the packed jar stand at room temp overnight. Once they have absorbed some of the oil, I return the jar to the refrigerator. The cold of the fridge will turn the oil cloudy but that is not a big deal. I then can use these by snipping them with my kitchen shears into salads, onto pizza (my personal favorite), put them on bruchetta, whatever you would use dried tomatoes for. I find that making my own results in a far superior product to what I might purchase at the market.