Lots of factors in play: mode of travel, length of stay, type of destination and available rentals at destination among others.
We've tried both approaches and while it is more comfortable to be on one's familiar steed, it is less of a travel headache to rent at the destination.
That said, what's available at the destination may be painful to ride and even to look at...
My thought is that every situation will be different. For example, we've had good experiences renting at the Gulf Coast of Alabama, but where we stay in the mountains there are no rentals to be had. Others will likely have had similar experiences.
So to quote the Washington Parish Fair, Cajun carnival barker, "Hey mis'ta, you pays you money and takes you chances!"
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Current bikes: Unknown year Specialized (rigid F & R) Hardrock, '80's era Cannondale police bike; '03 Schwinn mongrel MTB; '03 Specialized Hard Rock (the wife's)
Gone away: '97 Diamondback Topanga SE, '97 Giant ATX 840 project bike; '01 Giant TCR1 SL; and a truckload of miscellaneous bikes used up by the kids and grand-kids
Status quo is the mental bastion of the intellectually lethargic...