The properties of a material depend on how its atoms are arranged, and predicting these arrangements from first principles is a longstanding challenge. Orbital-free density functional theory provides a quantum-mechanical model based solely on the electron density, not individual wave functions. The resulting speedups make it attractive for random structure searching, whereby random configurations of atoms are relaxed to local minima in the energy landscape. We use this strategy to map the low-energy crystal structures of Li, Na, Mg, and Al at zero pressure. For Li and Na, our searching finds numerous close-packed polytypes of almost-equal energy, consistent with previous efforts to understand their low-temperature forms. For Mg and Al, the searching identifies the expected ground state structures unambiguously, in addition to revealing other low-energy structures. This new role for orbital-free density functional theory particularly as continued advances make it accurate for more of the periodic table-will expedite crystal structure prediction over wide ranges of compositions and pressures.