Not at all! I use a "gravel bike", which is sort of made for this kind of stuff (slightly relaxed road riding position, more aggressive weight balance towards rear, wide tires, beefier frame, plenty hardpoints for water). A road bike would suffice with wide enough tires (probably at least 38mm). The surface is mostly packed pea gravel, but there is some clay dirt streches and some spots that are "Kansas Road" grade... meaning chipped/rutted chert/limestone.

A mountain bike would certainly work but is overkill. The upright riding position of a mountain bike catches a lot of wind and you'll expend a lot of calories/water inefficiently, and they're not made for low rolling resistance.