There's a new startup that's servicing downtown: Memphis Pedicab Company. The intent is to provide downtown workers an easier way to travel across downtown in a door-to-door fashion. They're posting pictures on their facebook and the people look like they've just won Cash Cab! That may be because the rides are FREE! The drivers work for tips!
There has been a pedicab company in Memphis before, as Paul Ryburn remembers and this Daily News article from 1998 but per my emails with MPC (this might not work: MPC is my abbreviation for Memphis Pizza Cafe.) it is not the same company. I'm with Paul in hoping that they stick around.
Think you'd like to try your pedal mettle? Contact MPC through this craigslist ad: http://memphis.craigslist.org/lab/2630237057.html.