We may have to wait a while - Voyager relied on a rare once every 175 year planetary alignment for gravitational slingshotting. It's only a shame the whole Grand Tour that Voyager had once been intended to be a small part of never happened.
Depending on the mission there's no doubt other alignments that would give the gravity assist.
Do we have any newer propulsion methods that would make a difference in these super-long-running missions? (I am not a rocket scientist... but I'm thinking along the lines of something like a low-impulse ion thruster that uses a relatively small amount of fuel but can run for a very, very long time)
Yes, the new missions now have a Delta-V of something like 11 kilometers per second, which is insane. For instance, the New Horizons which only took 9 years to get to Pluto had thrusters with a Delta-V of only like 0.27 kilometers per second, so most of the trip was done using gravity assists. Missions now can do some crazy things like orbit two separate bodies.
To put it into perspective, Voyager 1 has a current speed of 17 kilometers per second. So the Dawn craft has most of that trip covered without gravity assists if it wanted. AFAIK there's nothing stopping us from loading up some craft with more fuel and larger ion engines except for launch weight. And even then, if we mastered orbital refueling we could surpass that.
As it stands, the craft can carry approximately the same amount of Delta-V that the rocket that got it into space. Equivalent to tons of fuel from chemical rockets.
> To put it into perspective, Voyager 1 has a current speed of 17 kilometers per second
And the Earth has 30 km/s, yet we don't get Voyager "for free" and then some just from LEO. You can't compare speeds in different parts of the gravity well against each other because then you're ignoring the dv required to get from here to there.
Depending on the mission there's no doubt other alignments that would give the gravity assist.
https://en.wikipedia.org/wiki/Grand_Tour_program