A couple of great produce storage suggestions I've learned over the years: for fruit (and tomatoes) purchased a little hard or unripe, let soften at room temp for a couple of days until it feels and smells ripe, (in a paper bag will hasten it, with a banana will hasten it more due to natural ethylene gases, plus the bag keeps the fruit flies away), then put it in the fridge to keep it from spoiling. Also for cucumbers, a great tip: get one of those perforated plastic bags they sell fresh bread in, from the bakery (You can buy the bread or just ask for the bag). Cucs will keep a couple of weeks in one of these bags. For asparagus: stand it up in a tall container with an inch of water in the bottom. Do NOT wrap in plastic or the tips will molder: keeps for 10 days at least. Lastly, wrap celery completely in aluminum foil and put in a plastic bag...keeps for many weeks. I have started experimenting with wrapping cut fruit (mango halves, watermelon) in aluminum foil to avoid plastic wrap...so far so good. The foil can be carefully cleaned and used over and over, and then it can be recycled!

I learned this from my boyfriend's mom: To keep leafy herbs (basil, cilantro) fresh, put them in water (as if they were flowers), then cover them with a (reused!) plastic bag and a rubberband. They will stay fresh for a week or more in the fridge.
Do the same with leaf lettuce, but you don't need to cover it with plastic. If you cut the bottom part of the plant off and add water every once in a while, you can keep lettuce fresh for weeks. No, I'm not exaggerating - try it!

And you can wash those bags in the dishwasher with a handy accessory called Bag-E-Wash. Or rinse out by hand and dry them on a bag dryer. Keep them clean and don't throw them, reuse them!

I have found one solution is the bags that absorb ethylene gas produced by the fruit or vegetables.  These bags are plastic, but are reusable and washable (Debbie Meyer Green Bags is one brand.  Fruit and veggies keep for two weeks or more in these bags, vs. a few days in regular plastic.

Really, it's not necessary.  I belong to a CSA and each week receive tons of fresh vegetables. Lettuce can easily be stored in a bowl with a damp towel over the top.  I've kept lettuce fresh this way for as much as 2 weeks.  If it starts to look a little less than full peak I run a little cold water over it, let it drain a bit and then back in the bowl it goes!.

Fresh herbs store just fine in a glass of water without the plastic bag - just rinse the herbs (gently) under cold water if they seem a little under the weather.  

I've found that hardier veggies can live just fine without bags of any kind.  Actually, it can be very important to keep them dry to avoid rot. Do not wash these until you intend to use them.  

Another great way to keep greens fresh and non-slimy is to use a salad spinner (typically easy to find used at your local thrift store).  Greens will stay fresh for at least 10 days in one of these, even after being washed.  If you can't find a spinner, place a colander full of greens inside a bowl and cover.

I couldn't help but add that there are some fantastic resources available for those who need to preserve an abundance of produce.  I recommend the books "Preserving the Harvest" by Carol Costenbader and "Joy of Pickling" by Linda Ziedrich as two thorough, yet approachable resources.  I use both extensively to preserve the veggies grown in my garden and have been able to eat my own pickles, preserves, soups, and sauces through the winter--yum!  

Ethylene gas sipping green bags!  They work! I can keep parsley from my garden fresh for a couple of weeks. Yes - they are plastic, but I wash them and reuse and reuse and reuse and reuse. Eventually - they get recyled with other plastic bags. . .

Point is - the vegetables are preserved for your healthy eating longer. . .

I too get tons of veggies from my CSA.  For all the green leafy items, I prewash and tear for easy use.  Store in my glass pyrex bowls with a damp paper towel on top and then cover with the lid (which is plastic).  This works great.  I've had stuff in my fridge for more than two weeks.  Fruit, I leave on the counter unless I know it's going to be a while before I eat it...including tomatoes.  Beans, I prewash and sanp and also store in glass pyrex bowls.  This works great for me :)

I agree with the salad spinner.  When I bring fresh romaine, spinach and other fresh greens home I take it apart, rinse it thorougly in the spinner in water to which I have added several drops of 3% peroxide.  Then I spin thoroughly and drain off all water. Store in the refrigerator in the spinner and you will be amazed how much longer the fresh greens last while being fresh and crisp.  I also can, dry or freeze vegetables and fruits when in season.  Asparagus soup, carrot soup, rhubarb crunch, baked apples, pesto are fabulous in the middle of winter or when a work crunch normally would send many people to the fast food restaurant. Freeze all foods in glass jars for best results. Fruit and tomatoes should never be stored in the refrigerator.
Since I grow a garden I am regularly making soups when the swiss chard, spinach, tomaotes,are getting ahead of me. The rabbits and deer tend to ravage much of my garden unfortunately even though I live in a fairly large city. However, the garden is the best therapy after working all day.
My mother used to say "1/2 - 1 hour from garden to the table (or canning/freezer container). She was absolutely right.  

There is another method to store lettuce and other greens that uses no plastic or paper towels and doesn't tie up the salad spinner, colander, bowls, etc.  Rinse the greens to clean them of dirt, bugs or whatever. Leave the leaves whole - torn leaves will start to rot more quickly.  Shake the leaves dry to remove loose moisture but leave them damp and wrap them in a large, fairly thick damp cloth (folded sort of envelope style).  I have found lightweight bath towels or the double layers of a flannel pillowcase to be the right size and thickness to hold a full head of lettuce or bunch of greens that will stay moist for quite a while.  You can redampen the towel when it dries out.  Pull out the amount you need for whatever you're making and fold the rest back up in the cloth and store it in the refrigerator.

Salad keeps best - and crispiest - if you wash it, then store it wet in a plastic bag. You wouldn't believe how crunchy butter lettuce gets after a few hours, and it keeps for days.

For a lot of veggies, it really pays to cook them fresh, right when the box is delivered, and then store them in glass, ready to incorporate into your dishes. It makes dinner prep a lot faster and the veggies seem to keep more of their kick.

Force-ripening fruit tends to lead to mushiness. Never refrigerate strawberries whole! If they're going bad, wash, cut and mix with a little sugar, then refrigerate. Then, within a day, have strawberry shortcake.

Just a quick comment: I buy organic potatoes that are not treated with whatever sprout-stifling chemical they use on conventional spuds. This means that unless they are kept in cold storage, the potatoes start to sprout and go soft within a week. And since most people don't use even the small 5 pound bag that quickly (unless you're planning on eating potatoes with every meal, which some might do), it really is best to refrigerate them. They take up about a quarter of my bottom shelf, but they last SO much longer, which is easier on the pocketbook AND cuts down on waste.

Plus, you can plant them in your garden if you want!

And here's a fun fact about the farmer I buy them from: instead of spending energy on refridgeration or a/c for his stored potatoes, this guy gets snow that is shoveled up and carted away from city streets and parking lots in the winter and packs it into these big metal boxes where it melts and freezes into huge ice blocks that keep his warehouse cool in the summer. Now how's that for energy efficient?!

are reusable, unbleached-cotton bags.  They can be mashine-washed and last much longer than plastic bags. They're good for veggies, grains, beans, and nuts.  Just not beets.  Unless you want red-splotched eco-bags.