The different color harmonies each have there own principles and function...
Complementary colors, when put together will excite the eye and really grab your attention due to their contrast. They can often have a nervous or vibrating effect. I personally hold great disdain for the harsh contrast between the colors purple and yellow. The only time I ever use them together is when I make softly blended filters. Together I think they work well in a winter theme. While yellow is a warm color, put it next to purple and I think its sharp and ice cold.
These colors are evenly spaced on the color wheel and therefor don't have a stark contrast that attracts the eyes. This scheme can be great for seasonal palettes, and the affect you will get is often very clean, simple, and soft. I'm a fan all the way around the wheel.
The analogous colors consist of one color that is contrast by two colors evenly spaced from its complementary. As apposed to the contrast found in the triadic colors they are much friendlier on the eyes and softer. I can see these color combinations as being easier to theme.
These colors can really grab your eye in a similar manner to the complementary colors. They are any three evenly spaced colors around the wheel. To me the colors blue, red, and yellow in their pure form seem kiddish. I find the combinations with two colors sharing a similar primary to be very attractive.
Square and Rectangle-
The square and rectangle are both double complementary. They consist of two sets of complementary colors, and the square's are evenly spaced. The square to me seems more playful while the rectangle seems more sophisticated. The danger of the double complementaries is that the closer you get to the primaries the more daunting they can get. It may be better to stagger the colors picking one to be supported by the others.