Laser-plasma accelerators have become compact sources of ultrashort electron bunches at energies up to the gigaelectronvolt range thanks to the remarkable progress made over the past decade. A direct application of these electron bunches is the production of short pulse x-ray radiation sources. In this letter, we study a fully optically driven x-ray source based on the combination of a laser-plasma accelerator and a plasma wave undulator. The longitudinal electric field of a laser-generated plasma wave is used to wiggle electrons transversally. The period of this plasma undulator being equal to the plasma wavelength, tunable photon energies in the 10 keV range can be achieved with electron energies in the 100–200 MeV range. Considering a 10s TW class femtosecond laser system, undulators with a strength parameter K ~ 0.5 and with about ten periods can be combined with a laser-plasma accelerator, resulting in several 10^-2 emitted x-ray photons per electron.