What kind of potatoes are you growing?

I see you live in Texas, it is much warmer down there.
If the flowers are falling off, I wouldn't worry at all, unless you are attempting to harvest seeds as well as a fresh crop of potatoes.

You plants will flower, and bud at the same time as all of your other tubers.
The flowers falling off are probably from a little too much water, but I wouldn't worry about it, it will have no affect whatsoever on your yield.

Are you positive that it is the flowers falling off, and NOT the buds???
That's odd.

It seems early for that.
The flowers will fall off if they are being over pollinated.
Lots of bes and bugs will knock off the flowers, that's not a problem and usually just indicated a great deal of moisture that year.

I'm not sure what would make the buds (eyes) fall off,
What type are you growing? I can find some information for you.
Russet potatoes

I'm checking.
I e-mailed a friend of mine, he is a professor at K-State. I figure an agricultural professor at one of the best agricultural and veterenary universities on the face of the earth should know a thing or two!! :D

I'll let you know as soon as I hear something back.