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petecldThat isn't caused by a cheap scanner. That effect is caused by the dot pattern used to print the cards. All scanners will pick it up. In your scanning program look for a option called "descreen" or "descreening". If you get an option asking you to pick a "DPI" or a "LPI" use 85 for anewspaper photo, 110 or 133 for magazines or cards. You'll notice it'll take a little longer to do a scan but it should rid you of that problem.
Yes, you can you the "blur" tool to get rid of it but then you have a blurry image. I suggest one use the despeckle filter to lessen this problem.