$L(M) = L$ where $M$ is a $TM$ that moves only to the right side so $L$ is regular
Hint -- You need to show that your TM has the same power as a finite-state automaton (as the commenter Dave Clarke said), that is, given such a TM, construct a FSA that accepts the same language.  

But since the TM has no memory but the tape, ask yourself what a right-only TM can do with its tape.  It should be relatively straightforward to actually construct the parts of the FSA you are looking for. Just go through them -- the states, the input alphabet and most crucially the move function (usually $\delta$) and define them in terms of the parts of the TM you started with. Then you have to show that the two accept the same language, in terms of the definition of "accept" for each, being sure to mention potential looping behavior in the TM.

BTW, this sort of problem is amenable to a pretty convincing "hand-waving" proof that would actually be of a type that is quite acceptable in a research paper.  Your course, however, may be expecting a precise proof, using $\delta$ and the other components of the tuples that constitute the TM and FSA.