Using self-consistent magnetohydrodynamic (MHD) simulations, we explore the hypothesis that nonlinear MHD waves dominate the internal dynamics of galactic molecular clouds. We employ an isothermal equation of state and allow for self-gravity. We adopt "slab-symmetry," which permits motions v_ and fields B_ perpendicular to the mean field, but permits gradients only parallel to the mean field. The Alfvén speed v_A exceeds the sound speed c_s by a factor 3-30. We simulate the free decay of a

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... rum of Alfvén waves, with and without self-gravity. We also perform simulations with and without self-gravity that include small-scale stochastic forcing. Our major results are as follows: (1) We confirm that fluctuating transverse fields inhibit the mean-field collapse of clouds when the energy in Alfvén- like disturbances remains comparable to the cloud's gravitational binding energy. (2) We characterize the turbulent energy spectrum and density structure in magnetically-dominated clouds. The spectra evolve to approximately v_, k^2≈ B_, k^2/4πρ∝ k^-s with s∼ 2, i.e. approximately consistent with a "linewidth-size" relation σ_v(R) ∝ R^1/2. The simulations show large density contrasts, with high density regions confined in part by the fluctuating magnetic fields. (3) We evaluate the input power required to offset dissipation through shocks, as a function of c_s/v_A, the velocity dispersion σ_v, and the scale λ of the forcing. In equilibrium, the volume dissipation rate is 5.5(c_s/v_a)^1/2 (λ/L)^-1/2×ρσ_v^3/L, for a cloud of linear size L and density ρ. (4) Somewhat speculatively, we apply our results to a "typical" molecular cloud. The mechanical power input required