  We study field theories on spaces with additional compact noncommutative
dimensions. As an example, we study \phi^3 on R^{1,3}\times T^{2}_\theta using
perturbation theory. The infrared divergences in the noncompact theory give
rise to unusual dynamics for the mode of \phi which is constant along the
torus. Correlation functions involving this mode vanish. Moreover, we show that
the spectrum of Kaluza-Klein excitations can be very different from the
analogous commuting theory. There is an additional contribution to the
Kaluza-Klein mass formula that resembles the contribution of winding states in
string theory. We also consider the effect of noncommutativity on the four
dimensional Kaluza-Klein excitations of a six dimensional gauge field.
