I think they do - I know our dog barks when he's fast asleep and we think its really funny - like humans talking in their sleep - which is related to dreaming
Certainly my dog twitches her paws when she's asleep - maybe dreaming of running? She also whimpers sometimes, with a strange expression on her face and when she wakes up she takes a few minutes to get back to grips with reality. Very much like a human having a nightmare. I assume someone somewhere has got a PhD for looking at this.
It has been found that all mammals dream while sleeping. But it is during REM (rapid eye movement) sleep that these dreams become clearer, highly detailed and vivid.
The only two mammals that don't experience REM sleep are the spiny ant eater and the dolphin, however it is still beleived they dream, as we do in non REM sleep. These dreams aren't as vivid as REM sleep dreams.
We all dream, every night. If someone says they don't dream, they are just unable to recall their dreams.