In looking at a starting .Net Core project - the starting point does not come with a "Structure"; VIsual Studio projects usually establish a starting project structure, which feels familiar.
I found David Fowler (works on ASP.NET core team), in 2014, suggested a structure.
src - Main projects (the product code)
tests - Test projects
docs - Documentation stuff, markdown files, help files etc.
samples (optional) - Sample projects
lib - Things that can NEVER exist in a nuget package
artifacts - Build outputs go here. Doing a build.cmd/build.sh generates artifacts here (nupkgs, dlls, pdbs, etc.)
packages - NuGet packages
build - Build customizations (custom msbuild files/psake/fake/albacore/etc) scripts
build.cmd - Bootstrap the build for windows
build.sh - Bootstrap the build for *nix
global.json - ASP.NET vNext only