I used this solar system simulation to look for planet positions in an arc, more or less as shown in this image of the Grand Tour (provided by @userLTK).
When I entered Voyager 1's launch date in the simulation, I got this image:
Jupiter-Saturn-Uranus are spread around 135 degrees, Jupiter-Saturn-Neptune are spread around 180 degrees. I basically checked positions every few years, looking for those angles.
In 2074, Jupiter, Saturn, and Uranus look to be nicely aligned:
The window for a 3-planet flyby is probably several years wide (just like a Grand Tour window).
Those 3 planets will be in conjunction in 2080. Saturn and Uranus align around 2036 as well, but Jupiter's too far away for a 3-planet tour.
And right now there's a favorable position for Jupiter, Saturn and Neptune:
(Google's search results for '3-planet conjunction' and similar terms were useless, by the way; you end up knee-deep in astrology)
There are more permutations for 3-planet tours than the two shown here, I didn't check all of them.
Now, this is just a first-order approach. You'd have to use trajectory calculation software to get exact answers. I found some Matlab scripts, but I don't have access to Matlab...