I have been told that you can and should avoid getting your passport stamped when entering Israel. Can anyone tell me more about this? Which middle Eastern countries will not allow entry when you have an Israel stamp? How can you avoid getting the stamp? If I do get a stamp what are the consecuences?

This is the rumour. You may be able to avoid the stamp just by asking for a separate sheet of paper instead. If you cross a land border with Egypt or Jordan, though, this will be moot since those stamps will be a dead giveaway of your presence in Israel.

There's a pretty good number of countries worldwide will bar entry to those with Israeli stamps. From wiki:
* Iran
* Kuwait
* Lebanon
* Libya
* Pakistan
* Saudi Arabia
* Sudan
* Syria
* Yemen

I've also seen UAE (Dubai) on some lists, though apparently they never really enforced it much.

The consequences only come into play if you wish to visit one of the countries that blacklist you. You won't be allowed in, it's just that simple. You won't be arrested or anything, just sent packing. Since many of them are lovely countries, I would certainly suggest avoiding the stamp if you can.

Israel has begun to get much more suspicious about people that do not want their passport stamped. I was travelling from Amman to Jerusalem with a fellow that wanted to avoid the stamp, and he got hassled. Remember, by saying you don't want the stamp you are stating that you are planning to travel to a country that is an enemy of Israel.

If you are travelling on an american passport, the much better option is to request a second passport from the State Dept. The easiest way to do it is to get the passport at the US embassy in Tel Aviv. They are used to it, know why you want it, and will have a passport for you in 7 days.

If you are travelling overland avoiding Israeli stamps will not be enough. Syria is notoriously tough and if they see stamps for Taba (the border crossing with egypt), or from any of the jordanian land crossings OR if they see that you are trying to come in from Jordan and you don't have any stamps showing how you got into Jordan, they will send you away.

Lebanon is also very strict, if they see any evidence of travel to Israel, you may be turned away. This includes items in your bag with hebrew, old plane tickets, etc.

If you have a US passport that currently has a stamp from Israel and you plan to travel to a country that would deny you entry as a result, you may want to do as Jose P. suggested and get a second U.S. passport. You would need to complete the process as if you were applying for a new passport. However, in addition to all of the standard required documents for a new US passport, you would also need to include with your application, a signed statement explaining the the exact reason that you are applying for second passport.