Many foreigners in Thailand feel that it is safer leaving their passport in the hotel/home safe and simply carry a photocopy of the real thing. Many others don't even do that. That is probably sufficient if the "Boys in Brown" flag you down on the highway for a routine traffic stop, but it probably would NOT be sufficient if the Immigrations Police get involved, such as they often do in bar busts.

Recently (Feb 21, 2010) the Bangkok Post included a quote from Lt. Gen. Wuthi Liptapallop, the new head of the Immigration Bureau:
But there were be no let up on the requirement for all foreigners to carry their passport with them all times, he said. "This is for identification purposes. It is a law. Every Thai national must carry their ID card with them at all times, so why should foreigners be exempt? To carry a copy is not acceptable because it is impossible to see whether it is counterfeit or altered. Carrying a passport is not something we thought up in Thailand, but a normal regulation in many countries. I have been to many countries and carried my passport with me all the time."
source: Ridding Thailand of foreign criminals

And, yes, I always carry my passport with me when in any other country other than my home country (USA), including when in Thailand.

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