Is there any problem carrying an empty alcohol stove on a plane? I was going to check it but thought it looked like a bomb and did not wish to have them rummaging through my checked luggage and making a mess.

In theory, one can carry or check a backpacking stove as long as it is brand new or there is no fuel bottle or there is absolutely no fuel residue.

In practice, it depends entirely on the training (and maybe the mood) of the person who xrays your bags.

http://www.tsa.gov/travelers/airtravel/assistant/editorial_1187.shtm

I always check my backpacking gear through. I’ve only had one problem. But no questions came up about my soda can stove. Then I buy fuel at my destination. My problem came when the airline lost my fixed-length hiking poles. But they found them 3 months later .

-Barry

I've checked in and carried on alcohol stoves many times without trouble.These were open jet or chimney types, not penny stoves, though. If security is very paranoid, I'd definitely would put it into hold luggage.
Even a wood stove in carry-on bag went through OK (stored inside my pot)but this one was lucky!.
I quit using my Whisperlite the day some security people got my fuel bottle and pump (disassembled!) from my checked-in bag, leaving only a broken zipper and an official explanation note !