  We describe and execute a novel approach to observationally estimate the
lifetimes of giant molecular clouds (GMCs). We focus on the cloud population
between the two main spiral arms in M51 (the inter-arm region) where cloud
destruction via shear and star formation feedback dominates over formation
processes. By monitoring the change in GMC number densities and properties from
one side of the inter-arm to the other, we estimate the lifetime as a fraction
of the inter-arm travel time. We find that GMC lifetimes in M51's inter-arm are
finite and short, 20 to 30 Myr. Such short lifetimes suggest that cloud
evolution is influenced by environment, in which processes can disrupt GMCs
after a few free-fall times. Over most of the region under investigation shear
appears to regulate the lifetime. As the shear timescale increases with
galactocentric radius, we expect cloud destruction to switch primarily to star
formation feedback at larger radii. We identify a transition from shear- to
feedback-dominated disruption through a change in the behavior of the GMC
number density. The signature suggests that shear is more efficient at
completely dispersing clouds, whereas feedback transforms the population, e.g.
by fragmenting high mass clouds into lower mass pieces. Compared to the
characteristic timescale for molecular hydrogen in M51, our short lifetimes
suggest that gas can remain molecular while clouds disperse and reassemble. We
propose that galaxy dynamics regulates the cycling of molecular material from
diffuse to bound (and ultimately star-forming) objects, contributing to long
observed molecular depletion times in normal disk galaxies. We also speculate
that, in more extreme environments such as elliptical galaxies and concentrated
galaxy centers, star formation can be suppressed when the shear timescale
becomes so short that some clouds can not survive to collapse and form stars.
