I want to travel in Europe, staying for longer than a tight budget allows. Is it possible to find temp jobs?

I mean is it possible to travel on a budget and stay like for a few weeks in each city? (I hate that kind of tours that go: visit 20 capitals in 15 days, you just rush and enjoy nothing).
Some people say it's possible to survive teaching conversational English to private students (under the table, collecting cash) without actually breaking the law and hopefully having enough time to make friends and appreciate the culture more deeply(having just a tourist visa, no work permit).
Any advice on this please???

LIVE IN PG ACCOMMODATION

What a great opportunity to be able to travel Europe. Once your there, people welcome temporary borders from America and often are helpful in referring you to a friend or acquaintance in other cities. My friend was able to do the same on a tight budget and found that the Europeans were very attentive and most people charged less than reasonable rates for accommodations. A travel agent with knowledge in the area of working in Europe could most likely give you information regards to a work permit and visa. Good luck to you and have fun!!

Well I think it is a bit hard when you are on the fly like that. I have lived in 3 different countries other then my own (I am Canadian) and that has been the best way to see things. When I was younger I went to school in Germany for a year and traveled on breaks, school trips, and backpacked around after. I worked in Afghanistan, and got to travel with work (okay maybe I wouldn't recommend that but I got to travel a lot) and right now me and my husband are in Denmark on working holiday visas. It is possible depending on where you are from to get working holiday visas for various European countries. This is mostly for Canadians, Austrialians, and New Zealanders. I don't know if you are male or female although a male can do this too but my cousin and my stepsister have both worked in Europe as Au Pairs. My cousin did it for a year I think and my stepsister did it in Stockholm for 2 months and then traveled. Now the best is probably if you have a University degree and you go and get a TESL certification then you can work legally in many places. I think you can get when with out a degree but it is more difficult.