I enjoy mixing some gravel and dirt into my road rides, though I don’t have a specific “mountain” or “gravel” bike– just my all-around bike with 38mm tires.
A local example I’ve enjoyed several times in the past few years is Wildcat Creek Trail, which takes you from Wildcat Canyon Park in Richmond up to Tilden Park in the Berkeley hills. It’s a good local introduction to gravel roads (and nice and wide compared to a typical multi-use trail, avoiding conflict with walkers and dogs), and I’ve brought along friends on road bikes and they’ve been able to handle it:
A short loop from Berkeley that bails out when you reach Little Farm in Tilden is something like this, about 16mi and 1100’ of climbing:
Or, a nice slightly longer loop from Oakland with twice as much climbing (riding uphill to the Tilden Steam Trains, then downhill to the Claremont Hotel and Fournee bakery) is something like this, about 24mi and 2100’ of climbing:
For a bit of local history, you can also visit the ruins / former site of the Belgum Sanitarium just off this route…