How to do routing with material-ui@next MenuItem?

You can use Link as parent component for MenuItem:

  <MenuList>
    <Link to="/myRoute" style={{ textDecoration: 'none', display: 'block' }}>
      <MenuItem>
        go to my route
      </MenuItem>
    </Link>
    <Link to="/anotherRoute" style={{ textDecoration: 'none', display: 'block' }}>
      <MenuItem>
        go to another route
      </MenuItem>
    </Link>
  </MenuList>

you can now give the MenuItem a component prop like this:

<MenuItem component={Link} to='/myRoute'>
go to my route
</MenuItem>