I have heard people say that having a stamp from Israel in your passport can cause problems when trying to get into other countries.
Is there any truth to this statement? If so, which countries does it create issues with?
Yes, having an Israeli stamp in your passport can restrict your ability to enter some Islamic and Arab nations including:
What to do about it? Here are some options...
Don't worry about it -- If you aren't interested in travelling to any of these countries, it doesn't matter if you have an Israeli stamp. This is the simplest option.
Stamp separate paper - Israeli passport control is amenable to stamping a separate piece of paper rather than stamping your passport. However, be aware that this may not be sufficient for hiding your visit to Israel. For example, if you cross a border on land (e.g. Jordan or Egypt), you will have the stamp from the nation on the other side of the border crossing, so not having an Israeli stamp isn't fooling anyone.
Renew passport - Simply get a new passport after visiting Israel. Then you can travel on your new passport to any of the above mentioned nations. This is an especially good option if your passport is getting close to the expiry date and/or you don't have any immediate plans to visit the countries in question.
Second passport - Some nations (at least in Europe) will actually issue a second passport for the purpose of travelling to Israel. The US does not do this, however, it isn't hard to get a second US passport for other valid reasons (such as needing to travel while your passport is away for visa processing). If taking this options, it is advised that you do not travel with both passports. Do not take a passport with an Israeli stamp to one of the listed nations even if you do not present it to officials as it may be found if your possessions are searched.
It is probably best not to visit Israel and one of these arab nations on the same trip as it is harder to conceal the visit (see note above about land border crossings). Other evidence of visiting Israel may also be problematic. I've heard of receipt/stamps from a hostel in Israel as cause for ejection from an arab country.
Source: written from memory though I believe most of the info came from the Lonely Planet guidebook