Various possibilities, the most likely being:

= too much nitrogen. I'll leave it to someone else to comment on what you can do about that.

= lack of fertilization because of cold temperatures. Your plastic cover will help keep the plants a bit warmer at night. Do you know what your overnight temps are (with and/or without the cover?)

Tomatoes are self-pollinating, but often need insects, wind, or people to shake the blossoms. You can flick blossoms or shake stems during ideal temperatures (warm but less than 90º). If the high is going to be above 90º, shake the blossoms before the hottest part of the day. It might help a few more fruit set.

You don't say what the variety/varieties are; some normally tend to set more fruit than others do (varying, of course, with conditions). Some varieties seem to have a limit of how much fruit they can support at any given time. So some blossom-drop is to be expected (though many cherry-types exhibit very little of it). Of course, your situation doesn't seem to be normal. [I mention this because some people think they have a problem if every blossom doesn't set fruit.]

Some varieties also naturally have larger and thicker foliage than others. The same for thick stems.

In case you haven't already seen the Blossom-drop FAQ, it's worth the read:
http://faq.gardenweb.com/faq/lists/tomato/2000083030027695.html

Thank you Missingtheobvious for taking the time to help me out. Your info and the FAQ were very helpful.