It's very likely that the 787 would have less problems with turbulence than the DC-8 did.
The wings of the Boeing 787 are more flexible than the DC-8, and that flexibility will damp the immediate impact of turbulence. More important, the Boeing 787 has a gust alleviation system that reacts to turbulence by counteracting the induced accelerations using the control surfaces.
Note that the aircraft certification standards have changed a lot since the DC-8 was certified. Originally the focus was very much on the maximum g-forces the wing could sustain, but this is not a very good measure for turbulence. The g-forces encountered in turbulence are the result of a combination of the turbulence itself and the aero-elastic response of the aircraft. Using a more flexible wing and gust alleviation systems will result in a much smoother ride (less g-forces) than a traditional very rigid wing.
Currently the certification standards define, in addition to a required g-force, the characteristics of the turbulence that the aircraft has to sustain. For example, see the FAA's regulation for large aircraft, FAR Part 25, Sec 25.341.
This takes into account a range of characteristics of turbulence that can be encountered as well as the dynamic response of the aircraft to that turbulence.
Apart for aerodynamic and structural differences between the DC-8 and the Boeing 787, better understanding of (mountain wave) turbulence, better weather prediction, better reporting of turbulence by aircrew (simply because there are more flights) and better weather radars on aircraft have reduced the risks of encountering turbulence since the DC-8 era.