Not if it doesn't in any way resemble US currency. Hell, you could use Monopoly money, if you chose to, and someone was willing to accept it. All any transaction is is barter, anyway... trading something of value to one for something of value to another.
If it looks like US currency, though, it becomes counterfeiting, and yeah, that gets you three hots and a cot and one other "benefit", too [noparse]http://www.cafepress.com/texasbigbird.47037848[/noparse]
I think there were some towns up in the north east that did this as well. They were trying to avoid inflation and promote local products and services. I am not sure if they are still doing that or not.