Using direct numerical simulations of early universe hydromagnetic turbulence with energy densities of up to 10% of the radiation energy density, we show that gravitational waves (GWs) with energy densities of about 10^{-10} times the critical energy density of the Friedmann universe today were produced. Their characteristic strain today is found to be about 10^{-20} and should be observable with the Laser Interferometer Space Antenna (LISA) in the mHz range. The GWs have positive (negative) circular polarization if the magnetic field has positive (negative) magnetic helicity. The GW energy reaches a constant value after the turbulent energy (kinetic or magnetic) has reached its maximum. Compressive modes are found to produce about 10 times stronger GWs than solenoidal ones. After a fraction of the Hubble time, a new shallow low frequency tail develops, which is shown not to be an artifact of a finite domain size.