Or just 'modernism,' or either term with a capital M.
At least, that's what people told me in this recent discussion: related discussion:modernist literature
See also related discussion:klassische Moderne
It depends on what text you're talking about, though. 'Modernism' and 'modernist' refer specifically to the literary and artistic movement of that name in the early 20th century. 'The modern period' could be the same thing, or it could be broader: for example, the period of the modern novel (since the 18th-19th c.), or even the period of modern English (since the 17th century) as opposed to Old English or Middle English.
'The modern age' or 'the modern era' would be both broader and oriented more toward history and culture in general than toward literature in particular: for example, the world since the Enlightenment and industrialization, or even since the advent of the modern nation-state.