The resulting moire patterns you see are just the result of tightly spaced parallel, or near parallel, lines that "beat" to some visible sensitivity of your eye. Some people see moire patterns where other people do not.
Whatever you are using to reduce your pictures might just be doing a simple "throw every other pixel away" to reduce your pictures to half, or some multiple thereof. There are other reduction schemes around that may be more satisfactory for you, for your particular pictures. But it really all depends upon what your picture is of. With some pictures, and some reduction schemes, a moire pattern may be created where a simple reduction scheme would not have created a problem at all.
All I am saying is that with your pictures you should experiment with a few different tools or methods. See what works best for you, with those pictures.
At the risk of sounding like I am making money every time I say it (and I don't, I am in no way connected to this product) Irfanview is a very fine graphics program with a half dozen different reduction/resample filters, which may work for you. And if that isn't enough for you, there is the even more elaborate (and still FREE!) XnView. XnView has much more capability, but I am not as well versed with its finer features.
An experienced (amateur)photographer, I finally made the leap into digital. Happy with camera but never had a need for graphics software before. I have three graphics programs (only Roxio Media Creator 7.5 is current). Only the old Adobe PhotoAlbum (? - I'm not home to check right now) could reduce the large file pictures of my house to a size appropriate for sending w/ email (~640 x 480 )without creating really serious moire patterns on the clapboard siding and roof. Yet I never see anything about such in any review or discussions. I will be purchasing software now that I need it, but worry about this undiscussed problem. Unfortunately, Photoship CS is not under consideration for $$$ reasons. Any comments or discussion? Or help? Thanks