Type (copy) into Terminal:

sudo /Applications/TextEdit.app/Contents/MacOS/TextEdit /System/Library/CoreServices/SystemUIServer.app/Contents/Resources/English.lproj/Localizable.strings

After you type your password, TextEdit will come up and it will open a text file (Localizable.strings) with root editing powers. Search for these lines:

/* Format screencapture file names */
"%@ %@ at %@" = "%1$@ %2$@ at %3$@";

Between the quotes on the right side of the equals sign is the date and time addition. Just edit it, something like this:

/* Format screencapture file names */
"%@ %@ at %@" = "Screen Shot";

Make sure to not leave this empty! Using the above example, the first screen shot file will be named Screen Shot, and then Screen Shot 1, Screen Shot 2, etc.

If you are using a system language other than English, change the Terminal command as needed. For example, French.lproj, German.lproj, etc. When you are ready, use Save (not Save As) in TextEdit and quit. If you want to see the result immediately, restart the SystemUIServer in Terminal:

killall SystemUIServer

Note that I tested this only on Snow Leopard.