Moires patterns despite using mipmaps
Two things come to mind:


When generating your smaller mip map levels try to avoid using a simple 2x2 box filter because, though cheap and cheerful, they do a really poor job of removing high frequency information (that exceeds the Nyquist limit) as well as over filtering some of the lower frequency information you need to keep. (Also, as an aside, you need to perform the MIP map generation in linear space. If your source data is sRGB, you thus need to map to and from linear).  FWIW, In Williams' original paper on MIP mapping, he said he used a "box (Fourier) window" to generate the prefiltered levels, which would be a sinc function in image space.
Standard Trilinear filtering will also over- and under-filter parts of the texture. If you don't like the aliasing you may have to a) adjust the bias to increase the filtering (which will increase blurring) and/or b) try using anisotropic filtering.