Linear Vlasov dispersion theory for a homogeneous, collisionless electron-proton plasma with Maxwellian velocity distributions is used to examine the damping of Alfv¿n-cyclotron fluctuations. Fluctuations of sufficiently long wavelength are essentially undamped, but as k$parallel$, the wave vector component parallel to the background magnetic field Bo, reaches a characteristic dissipation value kd, the protons become cyclotron resonant and damping begins abruptly. For proton cyclotron damping, kdc/ωp ~ 1 for 10-3 $lesssim$ ¿p $lesssim$ 10-1, where ¿p ≡ 8πnpkBTp/Bo2 and ωp/c is the proton inertial length. At k$parallel$ < kd, me/mp < ¿e, and ¿p $lesssim$ 0.10 the electron Landau resonance becomes the primary contributor to fluctuation dissipation, yielding a damping rate that scales as ωr $sqrt beta_{e}$ (k$perp$c/ωp)2, where ωr is the real frequency and k$perp$ is the wave vector component perpendicular to Bo. As ¿p increases from 0.10 to 10, the proton Landau resonance makes an increasing contribution to damping of these waves at k$parallel$ < kd and 0¿ < $theta$ < 30¿, where $theta$ = arccos($hat {bf k}$ ¿ $hat {bf B}$o). The maximum damping rate due to the proton Landau resonance scales approximately as ¿p(kc/ωp)2 over 0.50 ≤ ¿p ≤ 10. Both magnetic transit time damping and electric Landau damping may contribute to Landau resonant dissipation; in the electron Landau resonance regime the former is important only at propagation almost parallel to Bo, whereas proton transit time damping can be relatively important at both quasi-parallel and quasi-perpendicular propagation of Alfv¿n-cyclotron fluctuations.