Why not an artifact? I mean, there is nothing that says the artifact has to do anything other than have the ability to bend time. So if your concern was to not give them an uber-magic item that they can do other things with, then, well, simply don't allow the artifact to do anything other than time travel.
Not arguing, just trying to understand.
Of course, artifact may seem to common a choice so perhaps that is why you said no to it?
No divine influence, and no artifacts. Well, that leaves ultra-powerful beings with uber-arcane abilities. Perhaps they have to find that one particular being to help them in each of the time periods. "Go back 30 years? yeah, if you want to get back home from the past, you'll have to find that same ancient dragon to send you back."
Or, how about some combination of things -
Ancient druid myth about how when the stars of of a proper alignment and (insert some other contingents) the person who stands right in the middle of the sacred grove of stones is sent to another time...
The other contingents could be complicated (like things they have to do in very distant locations and then still be back at the grove in time for the star alignment) or it could be less taxing like covering themselves in sacred oils while ancient druids surround them and chant in unison (using a secret spell that is 9th level that has to be cast by 10 druids at one time to have any effects).
Then, in these other time periods, they will have to find similar sacred druid sites AND wait for the proper star alignment in that time period ("What, we have only one week until the stars align? we have to move fast" or "what? they won't realign for 3 months? what will we do to kill time once we vanquish the bad guy in this time period"
)
Anyway, I'm rambling now. sorry. hope something in my rambles proved somewhat helpful.
