The microtubule cytoskeleton is essential in maintaining the shape, strength and organization of cells and its misregulation has been implicated in neurological disorders and cancers. As the stiffest biological polymers, microtubules are also an excellent model system for rigid rod polymer gels. To better understand the structure-mechanics relationships in microtubule networks, we measure the force-dependent viscoelastic responses of entangled and crosslinked microtubule networks subjected to precise microscale manipulation. We use magnetic tweezers devices to apply calibrated step stresses and measure the resultant strain as a function of time. At short times the material behaves as an elastic solid. The linear regime is large, with gentle stiffening observed in entangled networks above ~70% strains. Crosslinked networks are stiffer, and the extent of linearity depends on the degree of crosslinking. In all cases, we find a creeping regime at long times, suggesting that structural rearrangements of the network dominate the mechanical response. To understand the molecular origins of this behavior, we use a newly-developed portable magnetic tweezers device to observe the network morphology using a confocal microscope while simultaneously applying point-like stresses to embedded magnetic particles. We observe substantial network compression in front of the bead with no evidence of long-length scale filament flow, and find that the spatial extent of the deformation field depends sensitively on network architecture and connectivity. Our results are important to understanding the role of the cytoskeleton in regulating cargo transport in vivo, as well as the basic physics of non- affine deformations in rigid rod polymer networks.