I started writing about a single example of complexity vs. simplicity here. Some continued thoughts...
Another related manner in which complexity adversely affects us is in the possession of "things." How much "stuff" do you own? How much clutter lies around you in your home? Here, too, Buddhism teaches simplicity. But before we get to that, let's just perform a simple thought experiment. Imagine yourself sitting in your home; everything is clean and nicely put away. Your decor is arranged as you like it, in an orderly fashion. You have the minimum number of knickknacks for decor as you feel comfortable with. You surroundings are clean, orderly, and arranged nicely. Take a moment to feel yourself in this scenario.
Now, take that scene and clutter it up. Add in extra knickknacks, extra "stuff." You can keep it orderly, but just add more and more. Whatever you can think to add, add it. Surround yourself with more and more decorations. Look around yourself in this new scene. Take a moment to feel yourself in this scenario. How do your feelings compare to the previous scene?
In Buddhism, renunciation plays a vital role. Complete renunciation is to become a monk. But even as lay practitioners, we are counselled to commit to renunciation. There are several aspects to this practice, and the one of particular interest to us here is that of renouncing sensual pleasures, one of which is worldly goods. As lay practitioners, we cannot, and should not, give up all of our possessions as do monks because we need more to live in society. However, we should carefully examine our possessions. Do we really need 3 television sets? 18 pairs of shoes? Are we really interested in that new CD because of the brilliance of the artist, or is it destined to become another coaster?
The Buddha instructed us in this manner for three reasons. The first is to combat attachment to things. We should be able to give away our goods when they are no longer needed, without feeling undue attachment. If we cannot, this teaches us where to focus our practice. The second is to combat our attachment to self. Owning more leads to the concept of "ownership" arising in our thoughts more often, which leads to the assumption that "I" am the owner, which further cements the delusional view of self that entraps us. The third is the most pragmatic—more stuff makes you feel less calm and clutters your mind with thoughts of that stuff. The image of a zendo, a zen meditation hall, opened this post. Minimal, simple decor relaxes the mind, allows it to focus more directly, more calmly, on seeing one's true nature—the purpose of Buddhist practice.
Subdue greed for sensual pleasures,
and see renunciation as rest.
Let there be nothing grasped
or rejected by you.
Burn up what's before,
and have nothing for after.
If you don't grasp
at what's in between,
you will go about, calm.
One completely devoid of greed
for name & form, brahman,
by which he would go
under Mara's sway. 
 Jatukannin's Question, translated by Thanissaro Bhikkhu. http://www.accesstoinsight.org/tipitaka/kn/snp/snp.5.11.than.html#rest