What is the reasoning behind the Amish practice of avoiding technological advances?

My stepfather's parents are Amish. I grew up being babysat some by my stepfathers mother who was a member of an Amish community in Arthur, Illinois and some by a friends family who happened to be Amish and went to school with me. As Flimzy points out in one of his comments Amish communities are more restricted than Mennonites. Amish people are not introverts they just want more than anything to be unlike what someone outside their group represents. My grandmother's house and farm did have running water in the kitchen but as far as I remember not in the bathrooms. She had a refrigerator and for a while a television but however in her community the television was considered a sin and later it was removed because of social pressures. If I remember correctly a minister drove by at night and happened to see the artificial light of the television changing color in the window. The reasons they are like they are are not just cultural but also because they choose to interpret the Bibles orthopraxic commands to the best of their ability. The head coverings (1 Corinthians 11:6), the plainness of everything about them and their things (1 Peter 3:3,1 Timothy 2:9), their interest in agriculture (Mark 4:14), thier love of people (Mark 12:30), their practice of shunning unrepentant saints who sin (1 Corinthians 5:11). I agree in principle with what Ryan said but I think these verses needs to be added as well.

1Pe 1:17 And if ye call on the Father, who without respect of persons judgeth according to every man's work, pass the time of your sojourning here in fear:

2Ti 2:4 No man that warreth entangleth himself with the affairs of this life; that he may please him who hath chosen him to be a soldier.

2Th 2:15 Therefore, brethren, stand fast, and hold the traditions which ye have been taught, whether by word, or our epistle.

Some of the reasons why Amish people do not do things involving some technologies is that they are just passing time till heaven as the scriptures require us to do. In this manner they justify their lack of certain things as unnecessary to life on the earth, materialistic and something which might cause them to become entangled.

Also in a way similar to the way in which Catholics defend their traditions an Amish person would defend theirs as well. If the common tradition held by a group is not against the word or sinful it is continued.

For Mennonites and Amish, the issue is not about the technology itself, but about community. Community is of paramount importance, and a technology is accepted or not depending on its effect on the community. So, for example, cars are bad because they permit people to travel long distances on a regular basis, reducing the connection to the local community. Similarly for telephones and the internet. This also means that Amish will use technology when it is necessary and not disruptive to the community. Most Mennonites will have no problem with using a bus or plane, or even renting a car (if that is possible) if they absolutely need to travel somewhere, say to visit a sick relative. Many allow the use of the internet for business purposes, but not for personal purposes. Bicycles, which they frequently own, can be modern design and use the latest materials and technologies, since a better bike doesn't disrupt the community. I know a Mennonite family that owns a highly complex welding robot for their business but doesn't allow personal computer use.

The other main consideration is the concept of being "plain", which means not ostentatious and is a virtue. That is the explanation for the distinctive dress, and the reason behind the absence of modern fabrics, zippers and such like.

As has been mentioned, Amish and Mennonites vary widely in what technology is permitted. Some permit the use of cars; some permit only black cars with no chrome ('plain'); some permit internal combustion on the farm or business; some forbid rubber tyres on their horse buggies. A lot of Mennonites (myself included) fully embrace technology. But for those that don't, in the end, community is the driving force.